SOLVING SYSTEMS USING SUBSTITUTION

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Presentation transcript:

SOLVING SYSTEMS USING SUBSTITUTION

Substitution is an algebraic model that can be used to find the exact solution of a system of equation. What does substitution mean? Give examples of where we have used substitution in Math before this unit.

USING SUBSTITUTION What is the solution of the system? Use substitution. Check your answer by using your graphing calculator. y = 3x and x + y = - 32 Step 1: Because y = 3x, you can substitute 3x for y in x + y = - 32 x + y = - 32 Write the second equation. x + 3x = -32 Substitute 3x for y 4x = -32 Simplify x = - 8 Divide each side by 4 Step 2: Substitute – 8 for x in either equation and solve for y. y = 3x Write either equation. y = 3(- 8) = -24 Substitute – 8 for x and solve. The solution is (-8, - 24). Check by substituting (- 8, - 24) into each equation.

Let’s try one more together. What is the solution of the system? Use substitution. Check your answer by substituting your solution into each equation. Use your graphing calculator to check your answer. y = 2x + 7 and y = x - 1

With a partner, solve the following system using substitution With a partner, solve the following system using substitution. Show all work. Check your answer by graphing each equation on your graphing calculator. y = 3x x + y = 8

SOLVING FOR A VARIABLE AND USING SUBSTITUTION What is the solution of the system? Use substitution. 3y + 4x = 14 and -2x + y = -3 Step 1: Solve one of the equations for one of the variables. -2x + y = - 3 Write the second equation. -2x + 2x + y = -3 Add 2x to each side. y = 2x – 3 Simplify. Step 2: Substitute 2x – 3 for y in the other equation and solve for x. 3y + 4x = 14 Write the first equation. 3(2x – 3) + 4x = 14 Substitute 2x – 3 for y. Use parentheses. 6x– 9 + 4x = 14 Use the distributive property. 10x = 23 Add 9 to each side and simplify. x = 2.3 Divide each side by 10.

Step 3: Substitute 2.3 for x in either equation and solve for y. -2x + y = - 3 Write either equation. -2(2.3) + y = - 3 Substitute 2.3 for x. - 4. 6 + y = - 3 Simplify. y = 1.6 Add 4.6 to each side. The solution is (2.3, 1.6). Use your graphing calculator to check your answer.

Let’s try one more together. What is the solution of the system? Use substitution. Use your graphing calculator to check your answer. 6y + 5x = 8 x + 3y = - 7 Which variable did you solve for? Which equation did you use to solve for the variable?

With your partner, solve the system using substitution With your partner, solve the system using substitution. Check your answer by substituting and by using your graphing calculator. x + 3 = y 3x + 4y = 7

Work to solve these systems. Show all your work. Check your answer. y = - x – 2 3x + y = 12 y = 6x 2x + 3y = - 20 x = 2y + 7 x = y + 4

Time to write…… When is the substitution method a better method than graphing for solving a system of linear equations? For each system, tell which equation you would first use to solve for a variable in the first step of the substitution method. Explain your choice. a. -2x + y = -1 and 4x + 2y = 12 b. x = 2y and 6x – y = 1 Tell whether each statement is true or false. Explain. a. When solving a system using substitution, if you obtain an identity (an equation with infinitely many solutions), then the system has no solution. b. You cannot use substitution to solve a system that does not have a variable with a coefficient of 1 or – 1.