20 Solving Equations To solve ONE step equations
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20 Solving Equations To solve ONE step equations you must use the INVERSE OPERATION. OPPOSITE Operation Inverse Operation Addition Subtraction Multiplication Division EX: V – 10 = -9 +10 V=1 EX: -40 = -5 p -5 -5 P=8 OP: Subtraction IN OP: Addition OP: Multiplication IN OP: Division 21
22 Two Step Equations have two operations that need to be undone in the opposite order of PEMDAS to solve. (SADMEP) EX: 3 X – 4 = 11 +4 +4 3 X = 15 3 3 X=5 OP: Multi, Subtraction IN OP: Addition, Division Solving Equations with variables on both sides 23 Step 1 Combine like Terms (CLT) • we add constants together • and add terms with the same variable and degree (exponent) • Degree is the exponent on the variable 1 Solution Only one value makes this equation true NO Solution No value makes this equation true Infinitely Many Solutions any value makes this equation true
24 Solving Equations With Multiple Variables To solve equations with multiple variables do the following: 1. Multiply both sides by the LCD to clear out any fractions. (Every term should be multiplied by the LCD) 2. Simplify both sides. This will often mean clearing out parenthesis and the like terms. 3. Move all terms containing the variable you are solving for to one side and all terms that do not contain the variable to the opposite side. 4. Divide by the coefficient of the variable. So that the coefficient becomes 1. Systems of Equations 25 A system of equations consists of 2 (or more) equations. The solution to a System of Equations is the ordered pair (x, y) that satisfies both equations. Types of Solutions • One solution intersecting lines • No solution parallel lines (never intersect, same slope) • Infinitely many solutions same line!
26 Solving Systems of Equations by Graphing When solving a system by graphing the solution is the point of intersection. Solving Systems of Equations with Substitution Substitute- “replace, plug in” Y must equal -2 to satisfy both equations Substitute y = - 2 from the first equation to the other Graph each line, then find the point of intersection Intersection point x y (-3, -5) Solution!! -1 -3 = 3 x 3 3 x = -1 -1 Solve for x Solution is Substitute y=2 x into the other equation CLT 4�� − �� = − 4 Solve for x x = -2 1. 2. 3. 4. WITH CALCULATOR Plug equations into y 1 and y 2 GRAPH 2 ND TRACE, Option 5, then ENTER 3 times Solution will be the ordered pair at the bottom x = -3 y = -5 (-3, -5) To find y, substitute x = -2 into either equation. (-2, -4) ( -1, -2) 27
28 Solving a System by Elimination with + or - ? If you do not have “ x = “ or “ y = “ pick the easiest variable in the easiest equation and solve for it. Solve for x So our system becomes Substitute x = 7 – 3 y for the other equation 2(7 – 3 y) – 4 y = -6 Solve for y 14 – 6 y – 4 y = -6 14 – 10 y = -6 -14 -10 y = -20 -10 y=2 x=1 Solution is (1, 2) 29 Eliminate – “get rid of” To eliminate a variable the coefficient in each equation has to be the same. We can eliminate the y-variable by adding the equations together Solve for x x=4 Substitute x = 4 into either equation to solve for x – 3 y =7 y. 4 – 3 y = 7 -4 -4 -3 y = 3 -3 -3 y = -1 Solution is (4, -1)
29 Solving a System by Elimination with Multiplication We can eliminate the x-variable by subtracting the equations together. Solve for y Goal: Rewrite the system so you can eliminate a variable with addition Choose which variable to Eliminate. To eliminate the x-variable, multiply the top Equation by -2 to get -10 x. ( y=3 Substitute y = 3 into either equation to solve for x. 3 x – y =-9 3 x – 3 = -9 +3 +3 3 x = -6 3 3 x = -2 + Solution is (-2, 3) Multiply one eqn by -1 Then add the eqns +( 5 Plug in y to find x y=1 y=4 Substitute y = 4 into either equation to solve for x. Alternative to subtracting ( Solve for y Solution is ( -1, 1) 5 x + y = 9 5 x + 4 = 9 -4 -4 5 x = 5 5 5 x=1 Solution is (1, 4) 30
31 Setting up Systems of Equations Word Problems Choose which variable to eliminate. To eliminate the y-variable both equations must be multiplied by a number. HINT: Choose the least common multiple as the target coefficient LCM (4, 9) = 36 so multiply the top by 9 and to get 36 y and multiply the bottom by 4 to get -36 y ( ( + Solve for x x = -6 Substitute x = -6 into either equation to solve for y. 5 x + 4 y = -30 5(-6) + 4 y = -30 +4 y = -30 +30 4 y = 0 4 4 y=0 Solution is (-6, 0) 32 Quantity and Price TJMS is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The tool in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the price of a senior citizen ticket and the price of a child ticket. Two Numbers The difference of two numbers 3. Their sum is 13. Find the numbers
Two Numbers 2. 0 The sum of digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number? Wind or Air Current Flying to Kampala with a tailwind a plane averaged 158 km/h. On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. Mixture A lab needs to make 100 gallons of an 18% acid solution by mixing a 12% acid solution with a 20% solution. How many gallons of each solution are needed?
