Solve this system of equations using the substitution method.
[tex]
\begin{array}{c}
y=-2 x \\
-4 x-20=y \\
([?], \quad)
\end{array}
[/tex]
Answer (2)
Substitute the expression for y from the first equation into the second equation: − 4 x − 20 = − 2 x .
Solve for x : x = − 10 .
Substitute the value of x back into the first equation to find y : y = − 2 ( − 10 ) .
Solve for y : y = 20 . The solution is ( − 10 , 20 ) .
Explanation
Understanding the problem We are given a system of two equations:
y = − 2 x − 4 x − 20 = y
We need to solve this system using the substitution method.
Substitution Since we already have an expression for y in terms of x from the first equation, we can substitute this expression into the second equation:
− 4 x − 20 = − 2 x
Solving for x Now, we solve for x :
Add 4 x to both sides:
− 20 = − 2 x + 4 x
− 20 = 2 x
Divide both sides by 2:
x = − 10
Solving for y Now that we have the value of x , we can substitute it back into the first equation to find the value of y :
y = − 2 x
y = − 2 ( − 10 )
y = 20
Final Answer So, the solution to the system of equations is x = − 10 and y = 20 . We can write this as an ordered pair: ( − 10 , 20 ) .
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is y = 5 x + 1000 (where y is the total cost and x is the number of units produced) and its revenue function is y = 15 x , solving this system of equations will give the number of units the company needs to sell to break even. In this case, solving the system gives x = 100 and y = 1500 , meaning the company breaks even when it sells 100 units and its total cost and revenue are both $1500.
The solution to the system of equations is the ordered pair (-10, 20). We found this by substituting y from the first equation into the second and solving for x, then using that value to find y. Therefore, the coordinates of the solution are (-10, 20).
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