What is the solution to the system of equations?
$\left\{\begin{array}{l}
10 x+3 y=43 \\
-9 x-y=-20
\end{array}\right.$
A. $(11,1)$
B. $(1,11)$
C. $(-1,11)$
D. $(11,-1)$
Answer (1)
Multiply the second equation by 3 to eliminate y.
Add the modified second equation to the first equation to eliminate y and solve for x: x = 1 .
Substitute the value of x back into the second equation and solve for y: y = 11 .
The solution to the system of equations is ( 1 , 11 ) .
Explanation
Problem Analysis We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously. The given equations are:
Given Equations 10 x + 3 y = 43 − 9 x − y = − 20
Solution Strategy To solve this system, we can use the method of substitution or elimination. Let's use the elimination method. We can multiply the second equation by 3 to eliminate y .
Multiply Second Equation by 3 Multiplying the second equation by 3, we get: 3 ( − 9 x − y ) = 3 ( − 20 ) − 27 x − 3 y = − 60
Eliminate y Now, we can add the modified second equation to the first equation to eliminate y : ( 10 x + 3 y ) + ( − 27 x − 3 y ) = 43 + ( − 60 ) − 17 x = − 17
Solve for x Solving for x , we have: x = − 17 − 17 = 1
Substitute x into Second Equation Now that we have the value of x , we can substitute it back into either of the original equations to solve for y . Let's use the second equation: − 9 x − y = − 20 Substituting x = 1 , we get: − 9 ( 1 ) − y = − 20 − 9 − y = − 20
Solve for y Solving for y , we have: − y = − 20 + 9 − y = − 11 y = 11
Solution So, the solution to the system of equations is x = 1 and y = 11 . We can write this as an ordered pair ( 1 , 11 ) .
Verification Let's verify the solution by substituting x = 1 and y = 11 into both original equations.Equation 1: 10 x + 3 y = 43 10 ( 1 ) + 3 ( 11 ) = 10 + 33 = 43 Equation 2: − 9 x − y = − 20 − 9 ( 1 ) − ( 11 ) = − 9 − 11 = − 20 Both equations are satisfied, so the solution is correct.
Final Answer Therefore, the solution to the system of equations is ( 1 , 11 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, suppose a company wants to know how many units of a product they need to sell to cover their costs. They can set up a system of equations where one equation represents the total cost (fixed costs plus variable costs) and the other represents the total revenue (price per unit times the number of units sold). Solving this system will give them the number of units they need to sell to break even, where total cost equals total revenue. This helps in making informed business decisions.
