Lesson 2 7 Solving Systems of Linear Equations

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Lesson 2. 7 Solving Systems of Linear Equations by Graphing Concept: Solving systems of

Lesson 2. 7 Solving Systems of Linear Equations by Graphing Concept: Solving systems of equations in two variables. EQ: How can I manipulate equations to solve a system of equations? (REI 5 -6, 10 -11) Vocabulary: System of equations, Inconsistent, Consistent (independent/dependent) ‘In Common’ Ballad: http: //youtu. be/Br 7 qn 4 y. Lf-I ‘All I do is solve’ Rap: http: //youtu. be/1 q. HTmxla. ZWQ 1 2. 2. 2: Solving Systems of Linear Equations

Introduction The solution to a system of equations is the point or points that

Introduction The solution to a system of equations is the point or points that make both equations true. Systems of equations can have one solution, no solutions, or an infinite number of solutions. There are various methods to solving a system of equations. One is the graphing method. On a graph, the solution to a system of equations can be easily seen. The solution to the system is the point of intersection, the point at which two lines cross or meet. 2 2. 2. 2: Solving Systems of Linear Equations

Key Concepts, continued Intersecting Lines Parallel Lines Same Line One solution No solutions Infinitely

Key Concepts, continued Intersecting Lines Parallel Lines Same Line One solution No solutions Infinitely many solutions Consistent Independent Inconsistent Consistent Dependent 2. 2. 2: Solving Systems of Linear Equations 3

Guided Practice Example 1 Graph the system of equations. Then determine whether the system

Guided Practice Example 1 Graph the system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has a solution, name it. 4 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 1, continued 1. Solve each equation for y. The first equation

Guided Practice: Example 1, continued 1. Solve each equation for y. The first equation needs to be solved for y. 4 x – 6 y = 12 Original equation – 6 y = 12 – 4 x Subtract 4 x from both sides. Divide both sides by – 6. Write the equation in slopeintercept form (y = mx + b). The second equation (y = –x + 3) is already in slopeintercept form. 2. 2. 2: Solving Systems of Linear Equations 5

Guided Practice: Example 1, continued 2. Graph both equations using the slopeintercept method. The

Guided Practice: Example 1, continued 2. Graph both equations using the slopeintercept method. The y-intercept of is – 2. The slope is . The y-intercept of y = –x + 3 is 3. The slope is – 1. 6 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 1, continued 3. Observe the graph. The lines intersect at the

Guided Practice: Example 1, continued 3. Observe the graph. The lines intersect at the point (3, 0). This appears to be the solution to this system of equations. 7 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 1, continued To check, substitute (3, 0) into both original equations.

Guided Practice: Example 1, continued To check, substitute (3, 0) into both original equations. The result should be a true statement. 4 x – 6 y = 12 First equation in the system 4(3) – 6(0) = 12 Substitute (3, 0) for x and y. 12 – 0 = 12 Simplify. 12 = 12 This is a true statement. y = –x + 3 Second equation in the system (0) = –(3) + 3 Substitute (3, 0) for x and y. 0 = – 3 + 3 Simplify. 0 = 0 This is a true statement. 2. 2. 2: Solving Systems of Linear Equations 8

Guided Practice: Example 1, continued 4. The system has one solution, (3, 0). This

Guided Practice: Example 1, continued 4. The system has one solution, (3, 0). This system is consistent because it has at least one solution and it is independent because it only has 1 solution. ✔ 9 2. 2. 2: Solving Systems of Linear Equations

Guided Practice Example 2 Graph the system of equations. Then determine whether the system

Guided Practice Example 2 Graph the system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has a solution, name it. 10 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued 1. Solve each equation for y. 11 2. 2.

Guided Practice: Example 2, continued 1. Solve each equation for y. 11 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued 2. Graph both equations using the slopeintercept method. The

Guided Practice: Example 2, continued 2. Graph both equations using the slopeintercept method. The y-intercept of both equations is 1. The slope of both equations is 2. 12 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued 3. Observe the graph. The graphs of y =

Guided Practice: Example 2, continued 3. Observe the graph. The graphs of y = 2 x +1 and -8 x + 4 y = 4 are the same line. There are infinitely many solutions to this system of equations. 13 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued 14 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued 14 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued 4. The system has infinitely many solutions. This system

Guided Practice: Example 2, continued 4. The system has infinitely many solutions. This system is consistent because it has at least one solution and it is dependent ✔ because it has more than one solution. 15 2. 2. 2: Solving Systems of Linear Equations

Guided Practice Example 3 Graph the system of equations. Then determine whether the system

Guided Practice Example 3 Graph the system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has a solution, name it. 16 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 3, continued 1. Solve each equation for y. The first equation

Guided Practice: Example 3, continued 1. Solve each equation for y. The first equation needs to be solved for y. – 6 x + 2 y = 8 Original equation 2 y = 8 + 6 x Add 6 x to both sides. y = 4 + 3 x Divide both sides by 2. Write the equation in slopey = 3 x + 4 intercept form (y = mx + b). The second equation (y = 3 x – 5) is already in slopeintercept form. 17 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 3, continued 2. Graph both equations using the slopeintercept method. The

Guided Practice: Example 3, continued 2. Graph both equations using the slopeintercept method. The y-intercept of y = 3 x + 4 is 4. The slope is 3. The y-intercept of y = 3 x – 5 is – 5. The slope is 3. 18 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 3, continued 3. Observe the graph. The graphs of – 6

Guided Practice: Example 3, continued 3. Observe the graph. The graphs of – 6 x + 2 y = 8 and y = 3 x – 5 are parallel lines and never cross. 19 2. 2. 2: Solving Systems of Linear Equations

Guided Practice: Example 3, continued 4. The system has no solutions because there are

Guided Practice: Example 3, continued 4. The system has no solutions because there are no values for x and y that will make both equations true. This system is inconsistent because it has no solutions. ✔ 20 2. 2. 2: Solving Systems of Linear Equations