875 9.1 Systems of Linear Equations True statements result when the solution is substituted in both equations, confirming that the solution set is 511, 426. S Now Try Exercise 7. EXAMPLE 1 Solving a System (Substitution Method) Solve the system. 3x + 2y = 11 (1) -x + y = 3 (2) SOLUTION Begin by solving one of the equations for one of the variables. We choose equation (2) because it can easily be solved for y. -x + y = 3 (2) y = x + 3 Add x. (3) Now replace y with x + 3 in equation (1), and solve for x. 3x + 2y = 11 (1) 3x + 21x + 32 = 11 Let y = x + 3 in (1). 3x + 2x + 6 = 11 Distributive property 5x + 6 = 11 Combine like terms. 5x = 5 Subtract 6. x = 1 Divide by 5. Replace x with 1 in equation (3) to obtain y = x + 3 = 1 + 3 = 4. The solution of the system is the ordered pair 11, 42. Check this solution in both equations (1) and (2). Note the careful use of parentheses. 3x + 2y = 11 (1) 3112 + 2142≟11 11 = 11 ✓ True -x + y = 3 (2) -1 + 4≟3 3 = 3 ✓ True CHECK y1 = −1.5x + 5.5 y2 = x + 3 −10 −10 10 10 To solve the system in Example 1 graphically, solve both equations for y. 3x + 2y = 11 leads to y1 = -1.5x + 5.5. -x + y = 3 leads to y2 = x + 3. Graph both y1 and y2 in the standard window to find that their point of intersection is 11, 42. Elimination Method The elimination method for solving a system of two equations uses multiplication and addition to eliminate a variable from one equation. To eliminate a variable, the coefficients of that variable in the two equations must be additive inverses. We use properties of algebra to change the system to an equivalent system, one with the same solution set. The three transformations that produce an equivalent system are listed here. Transformations of a Linear System 1. Interchange any two equations of the system. 2. Multiply or divide any equation of the system by a nonzero real number. 3. Replace any equation of the system by the sum of that equation and a multiple of another equation in the system.
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