10 A 3 Mathematics Systems of linear equations

10 A 3 Mathematics ﺃﻨﻈﻤﺔ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﻭﺍﻟﻤﻨﺤﻨﻴﺎﺕ Systems of linear equations and Curves ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The equation of a circle One more graph that you should recognize is the graph of a circle centred on the origin. y We can find the relationship between the x and y-coordinates on this graph using Pythagoras’ theorem. (x, y) r 0 x y x Let’s call the radius of the circle r. We can form a right angled triangle with length y, height x and radius r for any point on the circle. Using Pythagoras’ theorem this gives us the equation of the circle as: x 2 + y 2 = r 2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Exploring the graph of a circle ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous equations Equations in two unknowns have an infinite number of solution y pairs. For example, x+y=3 3 x+y=3 We can represent the set of solutions on a graph: 0 x 3 Another equation in two unknowns will also have an infinite y number of solution pairs. For example, This set of solutions can also be represented in a graph: 3 y–x=1 0 3 x ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous equations There is one pair of values that solves both these equations: x+y=3 y–x=1 We can find the pair of values by drawing the lines x + y = 3 and y – x = 1 on the same graph. y y–x=1 The point where the two lines intersect gives us the solution to both equations. 3 0 3 x+y=3 x This is the point (1, 2). At this point x = 1 and y = 2. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous equations x+y=3 y–x=1 are called a pair of simultaneous equations. The values of x and y that solve both equations are x = 1 and y = 2, as we found by drawing graphs. We can check this solution by substituting these values into the original equations. 1+2=3 2– 1=1 Both the equations are satisfied and so the solution is correct. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Solving simultaneous equations graphically ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

One linear and one quadratic equation y We can demonstrate the solutions to y=x+3 y = x 2 + 1 y=x+3 using a graph. y = x 2 + 1 (2, 5) (– 1, 2) 0 x It is difficult to sketch a parabola accurately. For this reason, it is difficult to solve simultaneous equations with quadratic terms using graphs, particularly when the solutions are not integers. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using graphs to solve equations We can also show the solutions to y = x 2 + 1 y=x+3 using a graph. The points where the two graphs intersect give the solution to the pair of simultaneous equations. 10 y = x 2 + 1 y=x+3 8 6 (2, 5) 4 (– 1, 2) – 4 – 3 – 2 2 – 1 0 – 2 1 2 3 4 It is difficult to sketch a parabola accurately. For this reason, it is difficult to solve simultaneous equations with quadratic terms using graphs, particularly when the solutions are not exact. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

One linear and one quadratic equation Suppose one of the equations in a pair of simultaneous equations is linear and the other is a quadratic of the form y = ax 2 + bx + c. By considering the points where the graphs of the two equations might intersect we can see that there could be two, one or no pairs of solutions. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Linear and quadratic graphs ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using graphs to solve equations Solve the equation 2 x 2 – 5 = 3 x using graphs. We can do this by considering the left-hand side and the righthand side of the equation as two separate functions. 2 x 2 – 5 = 3 x y = 2 x 2 – 5 y = 3 x The points where these two functions intersect will give us the solutions to the equations. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using graphs to solve equations 10 The graphs of y = 3 x y = 2 x 2 – 5 and y = 3 x intersect at the points: y = 2 x 2 – 5 8 (2. 5, 7. 5) 6 and (2. 5, 7. 5). 4 2 – 4 – 3 – 2 – 1 0 – 2 (– 1, – 3) – 4 – 6 (– 1, – 3) 1 2 3 4 The x-value of these coordinates give us the solution to the equation 2 x 2 – 5 = 3 x as x = – 1 and x = 2. 5 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using graphs to solve equations Solve the equation 2 x 2 – 5 = 3 x using graphs. Alternatively, we can rearrange the equation so that all the terms are on the right-hand side, 2 x 2 – 3 x – 5 = 0 y = 2 x 2 – 3 x – 5 y=0 The line y = 0 is the x-axis. This means that the solutions to the equation 2 x 2 – 3 x – 5 = 0 can be found where the function y = 2 x 2 – 3 x – 5 intersects with the x-axis. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using graphs to solve equations 10 The graphs of y = 2 x 2 – 3 x – 5 and y = 0 intersect at the points: y = 2 x 2 – 3 x – 5 8 (– 1, 0) 6 and (2. 5, 0). 4 2 (2. 5, 0) (– 1, 0) – 4 – 3 – 2 – 1 0 – 2 – 4 – 6 1 2 y=0 3 4 The x-value of these coordinates give us the same solutions x = – 1 and x = 2. 5 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

One linear and one quadratic equation If the second equation contains terms in y 2 the shape of the corresponding graph will not be a parabola but a circle: A line and a circle Again we can have two, one or no pairs of solutions. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

One linear and one quadratic equation Demonstrating these solutions graphically gives: x 2 + y 2 = 13 y y=x+1 (2, 3) 0 x (– 3, – 2) The graphs intersect at the points (– 3, – 2) and (2, 3). ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Linear and circular graphs ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous equations with no solutions Sometimes pairs of simultaneous equations produce graphs that are parallel. Parallel lines never meet, and so there is no point of intersection. When two simultaneous equations produce graphs which are parallel there are no solutions. How can we tell whether the graphs of two lines are parallel without drawing them? Two lines are parallel if they have the same gradient. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous equations with no solutions We can find the gradient of the line given by a linear equation by rewriting it in the form y = mx + c. The value of the gradient is given by the value of m. Show that the simultaneous equations y – 2 x = 3 2 y = 4 x + 1 have no solutions. Rearranging these equations in the form y = mx + c gives, y = 2 x + 3 y = 2 x + ½ The gradient m is 2 for both equations and so there are no solutions. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous equations with infinite solutions Sometimes pairs of simultaneous equations are represented by the same graph. For example, 2 x + y = 3 6 x + 3 y = 9 Notice that each term in the second equation is 3 times the value of the corresponding term in the first equation. Both equations can be rearranged to give y = – 2 x + 3 When two simultaneous equations can be rearranged to give the same equation they have an infinite number of solutions. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Special solutions ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method If two equations are true for the same values, we can add or subtract them to give a third equation that is also true for the same values. For example, suppose 3 x + y = 9 5 x – y = 7 Adding these equations: 3 x + y = 9 + 5 x – y = 7 8 x divide both sides by 8, The y terms have been eliminated. = 16 x=2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method Adding the two equations eliminated the y terms and gave us a single equation in x. Solving this equation gave us the solution x = 2. To find the value of y when x = 2 substitute this value into one of the equations. 3 x + y = 9 5 x – y = 7 Substituting x = 2 into the first equation gives us: 3× 2+y=9 6+y=9 subtract 6 from both sides: y=3 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method We can check whether x = 2 and y = 3 solves both: 3 x + y = 9 5 x – y = 7 by substituting them into the second equation. 5× 2– 3=7 10 – 3 = 7 This is true, so we have confirmed that x=2 y=3 solves both equations. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method Solve these equations: 3 x + 7 y = 22 3 x + 4 y = 10 Subtracting gives: 3 x + 7 y = 22 – 3 x + 4 y = 10 3 y = 12 y=4 divide both sides by 3; The x terms have been eliminated. Substituting y = 4 into the first equation gives us, 3 x + 7 × 4 = 22 3 x + 28 = 22 3 x = – 6 subtract 28 from both sides: x = – 2 divide both sides by 3: ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method We can check whether x = – 2 and y = 4 solves both, 3 x + 7 y = 22 3 x + 4 y = 10 by substituting them into the second equation. 3 × – 2 + 7 × 4 = 22 – 6 + 28 = 22 This is true and so, x = – 2 y=4 solves both equations. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method 1 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method Sometimes we need to multiply one or both of the equations before we can eliminate one of the variables. For example, 4 x – y = 29 1 3 x + 2 y = 19 2 We need to have the same number in front of either the x or the y before adding or subtracting the equations. Call these equations 1 and 2. 2× 1 : 8 x – 2 y = 58 + 3 x + 2 y = 19 3 + 2 : 11 x divide both sides by 11: 3 = 77 x=7 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method To find the value of y when x = 7 substitute this value into one of the equations, 1 4 x – y = 29 3 x + 2 y = 19 2 Substituting x = 7 into 1 gives, 4 × 7 – y = 29 28 – y = 29 subtract 28 from both sides, multiply both sides by – 1, –y = 1 y = – 1 Check by substituting x = 7 and y = – 1 into 2 , 3 × 7 + 2 × – 1 = 9 21 – 2 = 19 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method Solve: 2 x – 5 y = 25 1 3 x + 4 y = 3 2 Call these equations 1 and 2. 3× 1 6 x – 15 y = 75 2× 2 – 6 x + 8 y = 6 3 – 4 , divide both sides by – 23, 3 4 – 23 y = 69 y = – 3 Substitute y = – 3 in 1 , 2 x – 5 × – 3 = 25 2 x + 15 = 25 2 x = 10 subtract 15 from both sides, x=5 divide both sides by 2, ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method 2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The elimination method ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Substitution When one equation in a system of equations is quadratic, we often solve them by substitution. Solve: y = x 2 + 1 y=x+3 1 2 Substituting equation 1 into equation 2 gives x 2 + 1 = x + 3 Collect all the terms onto the left-hand side so that we can factor and use the Zero Product Property: x 2 – x – 2 = 0 (x + 1)(x – 2) = 0 x = – 1 or x=2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Substituting into equations We can then substitute these values of x into one of the original equations: y = x 2 + 1 or y = x + 3. To find the corresponding values of y it may be easier to substitute into the linear equation. When x = – 1 we have: When x = 2 we have: y = – 1 + 3 y=2+3 y=2 y=5 The solutions for this set of simultaneous equations are: x = – 1, y = 2 and x = 2, y = 5. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Elimination We could also have solved this system of equations using the elimination method. Solve: Subtract equation 1 from equation 2 : y = x 2 + 1 y=x+3 1 2 0 = x 2 – x – 2 This is the same single quadratic equation as the one we found using the substitution method. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations When one of the equations in a pair of simultaneous equations is quadratic, we often end up with two pairs of solutions. For example, 1 y = x 2 + 1 2 y=x+3 Substituting equation 1 into equation 2 , x 2 + 1 = x + 3 We have to collect all the terms onto the left-hand side to give a quadratic equation of the form ax 2 + bx + c = 0. x 2 – x – 2 = 0 factorize: (x + 1)(x – 2) = 0 x = – 1 or x=2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations We can substitute these values of x into one of the equations y = x 2 + 1 y=x+3 1 2 to find the corresponding values of y. It is easiest to substitute into equation 2 because it is linear. When x = – 1 we have, When x = 2 we have, y = – 1 + 3 y=2+3 y=5 The solutions are x = – 1, y = 2 and x = 2, y = 5. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations e. g. 1 Since the y-values are equal we can eliminate y by equating the right hand sides of the equations: This is a quadratic equation, so we get zero on one side and try to factorise: To find the y-values, we use the linear equation, which in this example is equation (2) The points of intersection are (1, 1) and (-3, 9) ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations Sometimes we need to rearrange the linear equation before eliminating y e. g. 2 Rearranging (2) gives Eliminating y: or Substituting in (2 a): ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Exercise Find the points of intersections of the following curve and line The solution is on the next slide ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations Solution: Rearrange (2): Eliminate y: Substitute in (2 a): The points of intersection are ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Special Cases e. g. 1 Consider the following equations: The line and the curve don’t meet. Solving the equations simultaneously will not give any real solutions ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations Suppose we try to solve the equations: Eliminate y: Calculating the discriminant, we get: The quadratic equation has no real roots. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations e. g. 2 Eliminate y: The discriminant, The quadratic equation has equal roots. Solving The line is a tangent to the curve. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Exercises Decide whether the following pairs of lines and curves meet. If they do, find the point(s) of intersection. For each pair, sketch the curve and line. 1. 2. 3. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations Solutions 1. the line is a tangent to the curve ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations Solutions 2. there are 2 points of intersection ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and quadratic equations Solutions 3. there are NO points of intersection ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and Circular graph. Look at this pair of simultaneous equations: y=x+1 x 2 + y 2 = 13 1 2 What shape is the graph given by x 2 + y 2 = 13? The graph of x 2 + y 2 = 13 is a circular graph with its centre at the origin and a radius of √ 13. We can solve this pair of simultaneous equations algebraically using substitution. We can also plot the graphs of the equations and observe where they intersect. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and Circular graph. y=x+1 x 2 + y 2 = 13 1 2 Substituting equation 1 into equation 2 , x 2 + (x + 1)2 = 13 expand the bracket: x 2 + x + 1 = 13 2 x 2 + 2 x + 1 = 13 simplify: subtract 13 from both sides: 2 x 2 + 2 x – 12 = 0 divide both sides by 2: x 2 + x – 6 = 0 factorize: (x + 3)(x – 2) = 0 x = – 3 or x=2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Simultaneous linear and Circular graph. We can substitute these values of x into one of the equations y=x+1 x 2 + y 2 = 13 1 2 to find the corresponding values of y. It is easiest to substitute into equation 1 because it is linear. When x = – 3 we have, When x = 2 we have, y = – 3 + 1 y = – 2 y=2+1 y=3 The solutions are x = – 3, y = – 2 and x = 2, y = 3. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Mastered ﺗﻥ Solve the simultaneous equations : ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻦ ﺍﻵﻨ ﺗﻴﻦ ﺍﻟ ﺍﻟﻴﺘﻴﻦ Solve the simultaneous equations ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Example Dave hits a ball along a path with height h = – 16 t 2 + 15 t + 3 where h is the height in feet and t is the time in seconds since the ball was hit. By chance, the ball hits a balloon released by a child in the crowd at the same time. The balloon’s height is given by h = 3 t + 5. What height is the balloon when the ball hits it? by elimination: – factor and solve for the time: h = 3 t + 5 h = – 16 t 2 + 15 t + 3 0 = 16 t 2 – 12 t + 2 0 = (4 t – 1)(4 t – 2) t=¼ or t=½ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Solution What height is the balloon when the ball hits it? collision time is given by: 0 = (4 t – 1)(4 t – 2) t=¼ or t=½ substitute these values of t into either of the original equations to find h: h = 3 t + 5 h=3×¼+5 = 5. 75 feet or h=3×½+5 = 6. 25 feet The balloon was at a height of either 5. 75 or 6. 25 feet when the ball hit it. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using the discriminant Once we have written two equations as a single quadratic equation, ax 2 + bx + c = 0, we can find the discriminant, b 2 – 4 ac, to find how many times the line and the curve intersect and how many solutions the system has. ● When b 2 – 4 ac > 0, there are two distinct points of intersection. ● When b 2 – 4 ac = 0, there is one point of intersection (or two coincident points). The line is a tangent to the curve. ● When b 2 – 4 ac < 0, there are no points of intersection. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using the discriminant 1 Show that the line y – 4 x + 7 = 0 is a tangent to the curve y = x 2 – 2 x + 2. 2 Call these equations 1 and 2. rearrange 1 to isolate y: substitute this expression into 2 rearrange into the usual form: y = 4 x – 7 : 4 x – 7 = x 2 – 2 x + 2 x 2 – 6 x + 9 = 0 b 2 – 4 ac = (– 6)2 – 4(9) = 36 – 36 =0 b 2 – 4 ac = 0 and so the line is a tangent to the curve. find the discriminant: ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation