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Use with Ready Instruction Lesson 16 Dear Family, Your child is learning to solve systems of equations algebraically. A system of equations is a set of related equations. A system of linear equations can have one solution, no solution, or infinitely many solutions. y 8 7 One way to find the solution to a system is to graph each equation. The point where the lines meet is the solution. y5x13 6 5 4 The graph shows the solution of the system: 3 y 5 2x y5x13 1 y 5 2x 2 21 O 21 The solution is the point of intersection, (3, 6). 1 2 3 4 5 x Another way to solve a system of equations is by manipulating the equations algebraically, using either substitution or elimination. These are the steps for solving a system using substitution. • Rewrite one equation (if necessary) in a form that sets y equal to an expression involving x. • Substitute the expression for y into the other equation. • Solve that equation to find the value of the second variable, x. • Substitute the value of x into either original equation to solve for y. • Check your answer. Consider the following example: Solve this system of equations. y5x12 2y 5 2x 1 13 On the next page you will see how your child will use substitution and another method called elimination to solve this system. ©Curriculum Associates, LLC Copying is not permitted. Lesson 16 Solve Systems of Equations Algebraically 163 Solve Systems of Equations Algebraically: Sample Solution Solve this system of equations. y 5 x 1 2 2y 5 2x 1 13 One way: Use substitution. The first equation is y 5 x 1 2, so there’s no need to rewrite. Solve for x. Substitute x 1 2 for y in the second equation. 2(x 1 2) 5 2x 1 13 2x 1 4 5 2x 1 13 Multiply. 3x 1 4 5 13 Add x to both sides. 3x 5 9 Subtract 4 from both sides. x 5 3 Divide both sides by 3. Solve for y. y 5 (3) 1 2 5 5 Substitute 3 for x in the first equation. The solution to the system is (3, 5). Another way: Use elimination. The coefficients of x are 1 and 21. They are opposites, which means their sum is zero, so adding the equations will eliminate the x-terms. y 5 x 1 2 2y 5 2x 1 13 y 1 2y 5 x 2 x 1 2 1 13 Add the equations. 3y 5 15 Simplify. y 5 5 Divide both sides by 3. 5 5 x 1 2 Substitute 5 for y in the first equation. 3 5 x Solve for x. The solution to the system is (3, 5). Answer: Both methods show that the solution to the system is (3, 5). 164 Lesson 16 Solve Systems of Equations Algebraically ©Curriculum Associates, LLC Copying is not permitted.