Using the substitution method, what is the solution to the following system of equations?
[tex]4 x-y=4[/tex]
[tex]6 x+2 y=-1[/tex]
A. [tex](\frac{1}{2}, 2)[/tex]
B. [tex](\frac{1}{2},-2)[/tex]
C. [tex](2 \frac{1}{2})[/tex]
D. [tex](-2, \frac{1}{2})[/tex]
Answer (2)
The solution to the system of equations is ( 2 1 , − 2 ) . The substitution method was used to solve for y in terms of x , then substitute into the second equation to find the value of x , and finally calculate y . The correct answer is option B.
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Solve the first equation for y : y = 4 x − 4 .
Substitute this expression into the second equation: 6 x + 2 ( 4 x − 4 ) = − 1 .
Simplify and solve for x : x = 2 1 .
Substitute the value of x back into the equation for y : y = − 2 . The solution is ( 2 1 , − 2 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
4 x − y = 4
6 x + 2 y = − 1
We will solve this system using the substitution method.
Solve for y First, we solve the first equation for y in terms of x :
4 x − y = 4
Subtract 4 x from both sides:
− y = 4 − 4 x
Multiply both sides by − 1 :
y = 4 x − 4
Substitute into second equation Next, we substitute this expression for y into the second equation:
6 x + 2 y = − 1
6 x + 2 ( 4 x − 4 ) = − 1
Solve for x Now, we simplify and solve for x :
6 x + 8 x − 8 = − 1
Combine like terms:
14 x − 8 = − 1
Add 8 to both sides:
14 x = 7
Divide both sides by 14:
x = 14 7 = 2 1
Solve for y Now that we have the value of x , we substitute it back into the equation y = 4 x − 4 to find the value of y :
y = 4 ( 2 1 ) − 4
y = 2 − 4
y = − 2
Final Answer Therefore, the solution to the system of equations is x = 2 1 and y = − 2 . We can write this as an ordered pair: ( 2 1 , − 2 ) .
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, a company might use a system of equations to model its costs and revenues, and then solve the system to find the level of production at which its costs equal its revenues. This is a crucial concept in business and economics, helping companies make informed decisions about pricing, production, and resource allocation. Understanding how to solve systems of equations is therefore a valuable skill for anyone interested in these fields. The solution ( 2 1 , − 2 ) represents a specific point where two linear relationships intersect, providing valuable insights in various scenarios.
