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How to Solve a System of Equations by Algebraic Reasoning (a.k.a. Elimination, Linear Combination, and/or Substitution) Example 1 Solve by Using Substitution Use substitution to solve the system of equations. 2x – y = –5 3y – 5x = 14 Solve the first equation for y in terms of x. 2x – y = –5 –y = –5 – 2x y = 5 + 2x First equation Subtract 2x from each side. Multiply each side by –1. Substitute 5 + 2x for y in the second equation and solve for x. 3y – 5x = 14 3(5 + 2x) – 5x = 14 15 + 6x – 5x = 14 15 + x = 14 x = –1 Second equation Substitute 5 + 2x for y. Distributive Property Simplify. Subtract 15 from each side. Now, substitute the value for x in either original equation and solve for y. First equation 2x – y = -5 Replace x with –1. 2(–1) – y = -5 Simplify. –2 – y = -5 Add 2 to each side. –y = -3 Multiply each side by –1. y =3 The solution of the system is (–1, 3). Standardized Test EXAMPLE Example 2 Solve by Substitution GRIDDABLE Maryn bought chairs and desks for her employees. The price of the chairs were $95 each and the price of the desks were $325 each. She bought a total of 23 desks and chairs and spent $4025. How many chairs did she buy? Show your work. Read the Test Item You are asked to find the number of chairs. Solve the Test Item Step 1 Define variables and write the system of equations. Let x represent the number of chairs and y represent the number of desks. x + y = 23 The total number of desks and chairs was 23. The total price was $4025. 95x + 325y = 4025 Step 2 Solve one of the equations for one of the variables in terms of the other. Since the coefficient of y is 1 and you are asked to find the value of x, it makes sense to solve the first equation for y in terms of x. x + y = 23 y = 23 – x Step 3 First equation Subtract x from each side. Substitute 23 – x for y in the second equation. 95x + 325y = 4025 95x + 325(23 – x) = 4025 95x + 7475 – 325x = 4025 -230x + 7475 = 4025 -230x = -3450 x = 15 Second equation Substitute 23 - x for y. Distributive Property Simplify. Subtract 7475 from each side. Divide each side by -230. Step 4 Maryn bought 15 chairs for her employees. Record your answer in a bubble grid like the one at the right. 1 5 Example 3 Solve by Using Elimination Use the elimination method to solve the system of equations. 3p + 4q = 0 p – 4q = –8 The coefficients of q for the first equation and the second equation are additive inverses. This means that when you add them the result will be 0 and the variable q will be eliminated. 3p + 4q = (+) p – 4q = 4p = p= 0 -8 -8 -2 Add the equations. Divide each side by 4. Now find q by substituting –2 for p in either original equation. 3p + 4q = 0 3(–2) + 4q = 0 –6 + 4q = 0 4q = 6 q= 3 First equation Replace p with –2. Multiply. Add 6 to each side. Divide each side by 4 and simplify. 2 The solution is 2, 3 . 2 Example 4 Multiply, Then Use Elimination Use the elimination method to solve the system of equations. 2x + 3y = 6 5x – 5y = 65 Multiply the first equation by 5 and the second equation by 2. Then subtract the equations to eliminate the x variable. 2x + 3y = 6 5x – 5y = 65 Multiply by 5. Multiply by 2. 10x + 15y = 30 (–) 10x – 10y = 130 25y = –100 y = –4 Replace y with –4 and solve for x. 2x + 3y = 6 2x + 3(–4) = 6 2x – 12 = 6 2x = 18 x=9 First equation Replace y with –4. Multiply. Add 12 to each side. Divide each side by 2. The solution is (9, –4). Subtract the equations. Divide each side by 25. Example 5 Inconsistent System Use the elimination method to solve the system of equations. 4x – y = -7 1 2x - y = 3 2 Use multiplication to eliminate x 4x – y = -7 1 2x - y = 3 2 4x – y = -7 Multiply by 2. 4x – y = 6 0 = -1 Add the equations. Since there are no values of x and y that will make the equation 0 = -1 true, there are no solutions for this system of equations.