\left\{\begin{array}{l}11 x-9 y=2 \\ 13 x+5 y=-11\end{array}\right.
Answer (2)
Multiply the first equation by 5 and the second equation by 9 to prepare for eliminating y .
Add the modified equations to eliminate y and solve for x : x = 172 − 89 .
Substitute the value of x into one of the original equations and solve for y : y = 172 − 147 .
The solution to the system of equations is x = 172 − 89 , y = 172 − 147 .
Explanation
Understanding the Problem We are given a system of two linear equations in two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously.
Choosing a Solution Method We have the following system of equations:
{ 11 x − 9 y = 2 13 x + 5 y = − 11
We can solve this system using the method of elimination or substitution. Let's use the elimination method.
Eliminating y Multiply the first equation by 5 and the second equation by 9 to eliminate y :
{ 5 ( 11 x − 9 y ) = 5 ( 2 ) 9 ( 13 x + 5 y ) = 9 ( − 11 )
This gives us:
{ 55 x − 45 y = 10 117 x + 45 y = − 99
Solving for x Add the two equations to eliminate y :
( 55 x − 45 y ) + ( 117 x + 45 y ) = 10 + ( − 99 )
172 x = − 89
Now, solve for x :
x = 172 − 89
Solving for y Substitute the value of x into the first original equation to solve for y :
11 ( 172 − 89 ) − 9 y = 2
− 172 979 − 9 y = 2
− 9 y = 2 + 172 979
− 9 y = 172 344 + 172 979
− 9 y = 172 1323
y = 172 1323 ÷ − 9
y = 172 1323 × 9 − 1
y = 1548 − 1323
y = 516 − 441
y = 172 − 147
Stating the Solution Therefore, the solution to the system of equations is:
x = 172 − 89 , y = 172 − 147
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling traffic flow in a city. In this case, solving a system of linear equations helps us find the exact values of two variables that satisfy two given conditions, which can be applied to resource allocation, financial planning, and engineering design.
To solve the given system of equations, we used the elimination method by multiplying the first equation by 5 and the second by 9 to eliminate y . After doing so and solving for x and y , we found that the solution is x = 172 − 89 and y = 172 − 147 .
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