Solve the equations of each system by applying the reduction method and graph one of the systems on the Cartesian plane.
a. [tex]\left\{\begin{array}{r}8 x+6 y=36 \\ 10 x-16 y=6\end{array}\right.[/tex]
b. [tex]\left\{\begin{array}{l}6 x+8 y=34 \\ 10 x+6 y=40\end{array}\right.[/tex]
c. [tex]\left\{\begin{array}{l}6 x+9 y=36 \\ 3 x-3 y=3\end{array}\right.[/tex]
d. [tex]\left\{\begin{array}{l}20 x+28 y=200 \\ 20 x+24 y=80\end{array}\right.[/tex]
Answer (1)
Solve system a using the reduction method: x = 47 153 , y = 47 78 .
Solve system b using the reduction method: x = 11 29 , y = 11 25 .
Solve system c using the reduction method: x = 3 , y = 2 .
Solve system d using the reduction method: x = − 32 , y = 30 .
Graph system c to visualize the solution as the intersection of two lines.
Final Answers: System a: x = 47 153 , y = 47 78 ; System b: x = 11 29 , y = 11 25 ; System c: x = 3 , y = 2 ; System d: x = − 32 , y = 30 .
Explanation
Problem Overview We are given four systems of linear equations and asked to solve each using the reduction method. We also need to graph one of the systems.
Solving System a System a: { 8 x + 6 y = 36 10 x − 16 y = 6 Multiply the first equation by 5 and the second equation by -4: { 40 x + 30 y = 180 − 40 x + 64 y = − 24 Add the equations: 94 y = 156 y = 94 156 = 47 78 ≈ 1.6596 Substitute y back into the first original equation: 8 x + 6 ( 47 78 ) = 36 8 x = 36 − 47 468 = 47 1692 − 468 = 47 1224 x = 47 × 8 1224 = 94 306 = 47 153 ≈ 3.2553 So the solution for system a is x = 47 153 , y = 47 78 .
Solving System b System b: { 6 x + 8 y = 34 10 x + 6 y = 40 Multiply the first equation by -5 and the second equation by 3: { − 30 x − 40 y = − 170 30 x + 18 y = 120 Add the equations: − 22 y = − 50 y = 22 50 = 11 25 ≈ 2.2727 Substitute y back into the first original equation: 6 x + 8 ( 11 25 ) = 34 6 x = 34 − 11 200 = 11 374 − 200 = 11 174 x = 11 × 6 174 = 11 29 ≈ 2.6364 So the solution for system b is x = 11 29 , y = 11 25 .
Solving System c System c: { 6 x + 9 y = 36 3 x − 3 y = 3 Multiply the second equation by -2: { 6 x + 9 y = 36 − 6 x + 6 y = − 6 Add the equations: 15 y = 30 y = 2 Substitute y back into the second original equation: 3 x − 3 ( 2 ) = 3 3 x = 3 + 6 = 9 x = 3 So the solution for system c is x = 3 , y = 2 .
Solving System d System d: { 20 x + 28 y = 200 20 x + 24 y = 80 Multiply the second equation by -1: { 20 x + 28 y = 200 − 20 x − 24 y = − 80 Add the equations: 4 y = 120 y = 30 Substitute y back into the first original equation: 20 x + 28 ( 30 ) = 200 20 x = 200 − 840 = − 640 x = − 32 So the solution for system d is x = − 32 , y = 30 .
Graphing System c Let's graph system c: { 6 x + 9 y = 36 3 x − 3 y = 3 We can rewrite these equations in slope-intercept form: { y = − 9 6 x + 9 36 = − 3 2 x + 4 y = 3 3 x − 3 3 = x − 1 The intersection point of these two lines is ( 3 , 2 ) , which confirms our solution.
Final Solutions The solutions for the systems are: System a: x = 47 153 ≈ 3.2553 , y = 47 78 ≈ 1.6596 System b: x = 11 29 ≈ 2.6364 , y = 11 25 ≈ 2.2727 System c: x = 3 , y = 2 System d: x = − 32 , y = 30
Examples
Systems of linear equations are used in various real-world applications, such as determining the optimal mix of products in manufacturing, balancing chemical equations, and modeling supply and demand in economics. For instance, a company might use a system of equations to determine the number of units of two different products they need to produce to maximize profit, given constraints on resources like labor and materials. By solving the system, they can find the production levels that satisfy all constraints and optimize their profit.
