Solve the system for $x$ and $y$.
$\begin{array}{l}
-8 x-5 y=-1 \\
-8 x-y=-45
\end{array}$
A. $(12,-19)$
B. $(7-11)$
C. $(6,-3)$
D. $(2,3)$
Answer (2)
The solution to the system of equations is ( x , y ) = ( 7 , − 11 ) , corresponding to option B. We found y by eliminating x and substituting back to find x . This method effectively solves the system using the elimination technique.
;
Subtract Equation 1 from Equation 2 to eliminate x : 4 y = − 44 .
Solve for y : y = − 11 .
Substitute y = − 11 into Equation 2: − 8 x − ( − 11 ) = − 45 .
Solve for x : x = 7 . The solution is ( 7 , − 11 ) .
Explanation
Analyzing the Problem 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.
Equation 1: − 8 x − 5 y = − 1 Equation 2: − 8 x − y = − 45
We can solve this system using either the substitution or elimination method. Here, we will use the elimination method.
Eliminating x To eliminate x , we can subtract Equation 1 from Equation 2:
( − 8 x − y ) − ( − 8 x − 5 y ) = − 45 − ( − 1 )
Simplifying this, we get:
− 8 x − y + 8 x + 5 y = − 45 + 1
4 y = − 44
Solving for y Now, we can solve for y :
4 y = − 44
y = 4 − 44
y = − 11
Solving for x Next, we substitute the value of y back into either Equation 1 or Equation 2 to solve for x . Let's use Equation 2:
− 8 x − y = − 45
− 8 x − ( − 11 ) = − 45
− 8 x + 11 = − 45
− 8 x = − 45 − 11
− 8 x = − 56
x = − 8 − 56
x = 7
Stating the Solution Therefore, the solution to the system of equations is x = 7 and y = − 11 . We can write this as the ordered pair ( 7 , − 11 ) .
Checking the Solution Now, let's check our solution with the given options. We found that the solution is ( 7 , − 11 ) , which corresponds to option b.
Examples
Systems of equations are used in various real-world applications. For example, they can be used to model supply and demand in economics, where the intersection of the supply and demand curves represents the equilibrium price and quantity. In engineering, systems of equations can be used to analyze electrical circuits or structural systems. In computer graphics, they are used to perform transformations and projections of objects in 3D space. Understanding how to solve systems of equations is a fundamental skill in many fields.
