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Systems of Linear Equations in Two Variables Systems of Linear Equations and Their Solutions We have seen that all equations in the form Ax + By = C are straight lines when graphed. Two such equations, such as those listed above, are called a system of linear equations. A solution to a system of linear equations is an ordered pair that satisfies all equations in the system. For example, (3, 4) satisfies the system x+y=7 (3 + 4 is, indeed, 7.) x – y = -1 (3 – 4 is indeed, -1.) Thus, (3, 4) satisfies both equations and is a solution of the system. The solution can be described by saying that x = 3 and y = 4. The solution can also be described using set notation. The solution set to the system is {(3, 4)} - that is, the set consisting of the ordered pair (3, 4). Text Example Determine whether (4, -1) is a solution of the system x + 2y = 2 x – 2y = 6. Solution Because 4 is the x-coordinate and -1 is the y-coordinate of (4, -1), we replace x by 4 and y by -1. x + 2y = 2 x – 2y = 6 ? ? 4 + 2(-1) = 2 4 – 2(-1) = 6 ? ? 4 + (-2) = 2 4 – (-2) = 6 ? 2 = 2 true 4+2=6 6 = 6 true The pair (4, -1) satisfies both equations: It makes each equation true. Thus, the pair is a solution of the system. The solution set to the system is {(4, -1)}. Solving Linear Systems by Substitution • Solve either of the equations for one variable in terms of the other. (If one of the equations is already in this form, you can skip this step.) • Substitute the expression found in step 1 into the other equation. This will result in an equation in one variable. • Solve the equation obtained in step 2. • Back-substitute the value found in step 3 into the equation from step 1. Simplify and find the value of the remaining variable. • Check the proposed solution in both of the system's given equations. Text Example Solve by the substitution method: 5x – 4y = 9 x – 2y = -3. Solution Step 1 Solve either of the equations for one variable in terms of the other. We begin by isolating one of the variables in either of the equations. By solving for x in the second equation, which has a coefficient of 1, we can avoid fractions. x - 2y = -3 This is the second equation in the given system. x = 2y - 3 Solve for x by adding 2y to both sides. Step 2 Substitute the expression from step 1 into the other equation. We substitute 2y - 3 for x in the first equation. x = 2y – 3 5 x – 4y = 9 Text Example cont. Solve by the substitution method: 5x – 4y = 9 x – 2y = -3. Solution This gives us an equation in one variable, namely 5(2y - 3) - 4y = 9. The variable x has been eliminated. Step 3 Solve the resulting equation containing one variable. 5(2y – 3) – 4y = 9 This is the equation containing one variable. 10y – 15 – 4y = 9 Apply the distributive property. 6y – 15 = 9 Combine like terms. 6y = 24 Add 15 to both sides. y=4 Divide both sides by 6. Text Example cont. Solve by the substitution method: 5x – 4y = 9 x – 2y = -3. Solution Step 4 Back-substitute the obtained value into the equation from step 1. Now that we have the y-coordinate of the solution, we back-substitute 4 for y in the equation x = 2y – 3. x = 2y – 3 Use the equation obtained in step 1. x = 2 (4) – 3 Substitute 4 for y. x=8–3 Multiply. x=5 Subtract. With x = 5 and y = 4, the proposed solution is (5, 4). Step 5 Check. Take a moment to show that (5, 4) satisfies both given equations. The solution set is {(5, 4)}. Solving Linear Systems by Addition • If necessary, rewrite both equations in the form Ax + By = C. • If necessary, multiply either equation or both equations by appropriate nonzero numbers so that the sum of the x-coefficients or the sum of the ycoefficients is 0. • Add the equations in step 2. The sum is an equation in one variable. • Solve the equation from step 3. • Back-substitute the value obtained in step 4 into either of the given equations and solve for the other variable. • Check the solution in both of the original equations. Text Example Solve by the addition method: 2x = 7y - 17 5y = 17 - 3x. Solution Step 1 Rewrite both equations in the form Ax + By = C. We first arrange the system so that variable terms appear on the left and constants appear on the right. We obtain 2x - 7y = -17 3x + 5y = 17 Step 2 If necessary, multiply either equation or both equations by appropriate numbers so that the sum of the x-coefficients or the sum of the y-coefficients is 0. We can eliminate x or y. Let's eliminate x by multiplying the first equation by 3 and the second equation by -2. Solution 2x – 7y = -17 3x + 5y = 17 Steps 3 and 4 Text Example cont. Multiply by 3. Multiply by -2. 3•2x – 3•7y = 3(-17) -2•3x + (-2)5y = -2(17) 6x – 21y = -51 -6x – 10y = -34 Add the equations and solve for the remaining variable. 6x – 21y = -51 -6x – 10y = -34 -31y = -85 Add: -31y = -85 -31 -31 y = 85/31 Divide both sides by -31. Simplify. Step 5 Back-substitute and find the value for the other variable. Backsubstitution of 85/31 for y into either of the given equations results in cumbersome arithmetic. Instead, let's use the addition method on the given system in the form Ax + By = C to find the value for x. Thus, we eliminate y by multiplying the first equation by 5 and the second equation by 7. Solution 2x – 7y = -17 3x + 5y = 17 Text Example cont. Multiply by 5. Multiply by 7. 5•2x – 5•7y = 5(-17) 7•3x + 7•5y = 7(17) 10x – 35y = -85 21x + 35y = 119 Add: 31x x = 34 = 34/31 Step 6 Check. For this system, a calculator is helpful in showing the solution (34/31, 85/31) satisfies both equations. Consequently, the solution set is {(34/31, 85/31)}. The Number of Solutions to a System of Two Linear Equations The number of solutions to a system of two linear equations in two variables is given by one of the following. Number of Solutions Exactly one ordered-pair solution No solution Infinitely many solutions y y x Exactly one solution What This Means Graphically The two lines intersect at one point. The two lines are parallel. The two lines are identical. y x No Solution (parallel lines) x Infinitely many solutions (lines coincide) Example Solve the system 2x + 3y = 4 -4x - 6y = -1 Solution: 2 (2x + 3y = 4) -4x - 6y = -1 4x + 6y = 8 -4x - 6y = -1 0=7 No solution multiply the first equation by 2 Add the two equations Systems of Linear Equations in Two Variables