Solve the system of equations:
[tex]
\begin{array}{l}
y=7 x-53 \\
y=-4 x+35
\end{array}
[/tex]
A. (-3,-8)
B. (8,3)
C. (-8,3)
D. No solution
Answer (2)
Set the two equations equal to each other: 7 x − 53 = − 4 x + 35 .
Solve for x : 11 x = 88 , which gives x = 8 .
Substitute x = 8 into one of the equations to solve for y : y = − 4 ( 8 ) + 35 = 3 .
The solution to the system of equations is ( 8 , 3 ) .
Explanation
Problem Analysis We are given a system of two linear equations:
Equation 1: y = 7 x − 53 Equation 2: y = − 4 x + 35
Our goal is to find the values of x and y that satisfy both equations. We can solve this system by setting the two equations equal to each other.
Setting Equations Equal Setting the two equations equal to each other:
7 x − 53 = − 4 x + 35
Solving for x Now, we solve for x :
Add 4 x to both sides:
7 x + 4 x − 53 = 35
11 x − 53 = 35
Add 53 to both sides:
11 x = 35 + 53
11 x = 88
Divide by 11 :
x = 11 88
x = 8
Solving for y Substitute the value of x into either equation to solve for y . Let's use Equation 2:
y = − 4 x + 35
Substitute x = 8 :
y = − 4 ( 8 ) + 35
y = − 32 + 35
y = 3
Verification So, the solution is ( x , y ) = ( 8 , 3 ) .
Now, let's check if this solution satisfies both equations:
Equation 1: 3 = 7 ( 8 ) − 53 ⇒ 3 = 56 − 53 ⇒ 3 = 3 (Correct)
Equation 2: 3 = − 4 ( 8 ) + 35 ⇒ 3 = − 32 + 35 ⇒ 3 = 3 (Correct)
The solution ( 8 , 3 ) satisfies both equations.
Final Answer Therefore, the correct answer is ( 8 , 3 ) .
Examples
Systems of equations are used in various real-life situations. For instance, when planning a budget, you might have different income sources and expenses. Each income source or expense can be represented as an equation, and solving the system of equations helps you determine how to allocate your funds effectively. Another example is in physics, where systems of equations can describe the motion of objects under multiple forces. Solving these equations helps predict the trajectory and behavior of the objects.
The solution to the system of equations is (8, 3), which satisfies both equations. By substituting the value of x back into either equation, we confirmed that y equals 3. Therefore, the correct answer choice is B: (8, 3).
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