Learn to solve systems of equations.

A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.

Example 1: Identifying Solutions of a System of Equations Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 1. (1, 2) 5x + y = 7 x – 3y = 11 Substitute for x and y. 5(1) + 2 = 7 ? 1 – 3(2) = 11 ? 7 = 7  –5  11  The ordered pair (1, 2) is not a solution of the system of equations.

Example 2: Identifying Solutions of a System of Equations Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 2. (2, –3) 5x + y = 7 x – 3y = 11 5(2) + –3 = 7 ? Substitute for x and y. 2 – 3(–3) = 11 ? 7 = 7 11 = 11   The ordered pair (2, –3) is a solution of the system of equations.

Example 3 Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 3. (1, 2) 4x + y = 8 x – 4y = 12 Substitute for x and y. 4(1) + 2 = 8 ? 1 – 4(2) = 12 ? 6  8  –7  12  The ordered pair (1, 2) is not a solution of the system of equations.

Example 4 Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 4. (1, 4) 4x + y = 8 x – 4y = 12 Substitute for x and y. 4(1) + 4 = 8 ? 1 – 4(4) = 12 ? 8 = 8  –15  12  The ordered pair (1, 4) is not a solution of the system of equations.

When solving systems of equations, remember to find values for all of the variables. Helpful Hint

Example 5: Solving Systems of Equations y = x – 4 Solve the system of equations. y = 2x – 9 y = y y = x – 4 y = 2x – 9 x – 4 = 2x – 9 Solve the equation to find x. x – 4 = 2x – 9 – x – x Subtract x from both sides. –4 = x – 9 + 9 + 9 Add 9 to both sides. 5 = x

Example 5 Continued To find y, substitute 5 for x in one of the original equations. y = x – 4 = 5 – 4 = 1 The solution is (5, 1). Check: Substitute 5 for x and 1 for y in each equation. y = x – 4 y = 2x – 9 1 = 2(5) – 9 ? 1 = 5 – 4 ? 1 = 1  1 = 1 

Example 6 y = x – 5 Solve the system of equations. y = 2x – 8 y = y y = x – 5 y = 2x – 8 x – 5 = 2x – 8 Solve the equation to find x. x – 5 = 2x – 8 – x – x Subtract x from both sides. –5 = x – 8 + 8 + 8 Add 8 to both sides. 3 = x

Example 6 Continued To find y, substitute 3 for x in one of the original equations. y = x – 5 = 3 – 5 = –2 The solution is (3, –2). Check: Substitute 3 for x and –2 for y in each equation. y = x – 5 y = 2x – 8 –2 = 2(3) – 8 ? –2 = 3 – 5 ? –2 = –2  –2 = –2 

To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

You can choose either variable to solve for You can choose either variable to solve for. It is usually easiest to solve for a variable that has a coefficient of 1. Helpful Hint

Example 7: Solving Systems of Equations Solve the system of equations. 7. 3x – 3y = -3 2x + y = -5 Solve both equations for y. 3x – 3y = –3 2x + y = –5 –3x –3x –2x –2x –3y = –3 – 3x y = –5 – 2x –3 3x –3y = – y = 1 + x 1 + x = –5 – 2x

Example 10 Continued 1 + x = –5 – 2x Add 2x to both sides. + 2x + 2x 1 + 3x = –5 Subtract 1 from both sides. –1 –1 3x = –6 Divide both sides by 3. –6 3 3x = x = –2 y = 1 + x Substitute –2 for x. = 1 + –2 = –1 The solution is (–2, –1).

Example 11 Solve the system of equations. 11. x + y = 5 3x + y = –1 Solve both equations for y. x + y = 5 3x + y = –1 –x –x – 3x – 3x y = 5 – x y = –1 – 3x 5 – x = –1 – 3x Add x to both sides. + x + x 5 = –1 – 2x

Example 11 Continued 5 = –1 – 2x + 1 + 1 Add 1 to both sides. 6 = –2x –3 = x Divide both sides by –2. y = 5 – x = 5 – (–3) Substitute –3 for x. = 5 + 3 = 8 The solution is (–3, 8).

Example 12 Solve the system of equations. 12. x + y = –2 –3x + y = 2 Solve both equations for y. x + y = –2 –3x + y = 2 – x – x + 3x + 3x y = –2 – x y = 2 + 3x –2 – x = 2 + 3x

Example 12 Continued –2 – x = 2 + 3x Add x to both sides. + x + x –2 = 2 + 4x Subtract 2 from both sides. –2 –2 –4 = 4x Divide both sides by 4. –1 = x y = 2 + 3x Substitute –1 for x. = 2 + 3(–1) = –1 The solution is (–1, –1).

Lesson Review 1. Determine if the ordered pair (2, 4) is a solution of the system. y = 2x; y = –4x + 12 Solve each system of equations. 2. y = 2x + 1; y = 4x 3. 6x – y = –15; 2x + 3y = 5 4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers. yes ( , 2) 1 2 (–2,3) 15 and 8