What is the solution of the system of linear equations?
$\begin{array}{l}
-3 x+4 y=-18 \\
2 x-y=7
\end{array}$
A. $(-2,-3)$
B. $(-2,3)$
C. $(2,-3)$
D. $(2,3)$
Answer (2)
Express y in terms of x from the second equation: y = 2 x − 7 .
Substitute this expression into the first equation and solve for x : − 3 x + 4 ( 2 x − 7 ) = − 18 A rr x = 2 .
Substitute the value of x back into the equation for y : y = 2 ( 2 ) − 7 A rry = − 3 .
The solution to the system of equations is ( 2 , − 3 ) .
Explanation
Analyze the problem We are given a system of two linear equations: − 3 x + 4 y = − 18 2 x − y = 7 We need to find the values of x and y that satisfy both equations. We can use the substitution or elimination method to solve this system.
Solve for x Let's use the substitution method. From the second equation, we can express y in terms of x :
y = 2 x − 7 Now, substitute this expression for y into the first equation: − 3 x + 4 ( 2 x − 7 ) = − 18 Simplify and solve for x :
− 3 x + 8 x − 28 = − 18 5 x = 10 x = 2
Solve for y Now that we have the value of x , we can substitute it back into the equation y = 2 x − 7 to find the value of y :
y = 2 ( 2 ) − 7 y = 4 − 7 y = − 3
State the solution Therefore, the solution to the system of equations is ( x , y ) = ( 2 , − 3 ) .
Examples
Systems of linear equations are used in various fields, such as economics, engineering, and computer science. For example, in economics, they can be used to model the supply and demand of goods in a market. In engineering, they can be used to analyze the forces acting on a structure. In computer science, they can be used to solve problems in linear programming and optimization. Understanding how to solve systems of linear equations is a fundamental skill in many areas of study and can help you make informed decisions in real-world situations.
The solution to the system of equations is ( 2 , − 3 ) . This was found by substituting one equation into the other and solving for both x and y . Therefore, the correct option is C.
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