Solving system of equations by substitution

1. Solving system of equations by substitution

2. Key concepts • There are various methods to solving a system of equations. Two methods include the substitution methodand the elimination method. • The substitution method involves solving one of the equations for one of the variables and substituting that into the other equation. • Solutions to systems are written as an ordered pair, (x,y). This is where the lines would cross if graphed.

3. Key concepts continued • If the resulting solution is a true statement, such as 9 = 9, then the system has an infinite number of solutions. This is where the lines would coincide if graphed. • If the result is an untrue statement, such as 4 = 9, then the system has no solutions. This is where lines would be parallel if graphed. • Check your answer by substituting the x and y values back into the original equations. If the answer is correct, the equations will result in true statements.

4. Steps for substitution method • Step 1: Solve one of the equations for one of the variables in terms of the other variable. • Step 2: Substitute, or replace, the resulting expression into the other equation. • Step 3: Solve the equation for the second variable. • Step 4: Substitute the found value into either of the original equations to find the value of the other variable.

5. Example 1: • Step 1: Solve one of the equations for one of the variables in terms of the other variable. • It doesn’t matter which equation you choose, nor does it matter which variable you solve for. • Let’s solve for the variable y. Isolate y by subtracting x from both sides.

6. Step 2: Substitute, or replace, into the other equation, . • It helps to place parentheses around the expression you are substituting. • Step 3: Solve the equation for the second variable. Second equation of the system. Substitute for y. Distribute the negative over Simplify. Add 2 to both sides. Divide both sides by 2.

7. First equation of the system. • Step 4: Substitute the found value, (), into either of the original equations to find the value of the other variable. • The solution to the system of equations is (). If graphed, the lines would cross at (). Substitute for . Simplify. Subtract from both sides.

8. Example 2: • Step 1: Solve one of the equations for one of the variables in terms of the other variable. • It doesn’t matter which equation you choose, nor does it matter which variable you solve for. • Let’s solve for the variable y. Isolate by adding to both sides.

9. Step 2: Substitute, or replace, into the other equation, . • It helps to place parentheses around the expression you are substituting. • Step 3: Solve the equation for the second variable. Second equation of the system. Substitute for . Distribute the 4 through Simplify. Add 12 to both sides. Divide both sides by 5.

10. First equation of the system. • Step 4: Substitute the found value, (), into either of the original equations to find the value of the other variable. • The solution to the system of equations is (). If graphed, the lines would cross at (). Substitute for y. Simplify. Add 8 to both sides.

11. You try! 1) 2)

12. Real-World Applications Example 1: A shopper purchased 4 tables and 2 chairs for $200 and 2 tables and 7 chairs for $400. What is the cost of each table and each chair? Set up a system of equations:

13. Example 1 continued • Solve for x and y:

14. Real-World Applications Example 2: If the length of the rectangle is twice the width, and the perimeter of the rectangle is 30 cm, what is the length and width of the rectangle? Set up a system of equations:

15. Example 2 Continued • Solve for and :