Solve the system of equations given below.
$\begin{array}{l}
-5 x=y-5 \\
-2 y=-x-21
\end{array}$
A. $(10,16)$
B. $(-1,10)$
C. $(-1,-23)$
D. $(10,-45)$
Answer (1)
Rewrite the given equations in standard form.
Solve for y in terms of x using the first equation: y = − 5 x + 5 .
Substitute the expression for y into the second equation and solve for x : x = − 1 .
Substitute the value of x back into the expression for y to find y = 10 . The solution is ( − 1 , 10 ) .
Explanation
Analyze 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.
The given equations are:
Equation 1: − 5 x = y − 5 Equation 2: − 2 y = − x − 21
Rewrite the equations Let's rewrite the equations in standard form to make them easier to work with.
Equation 1: − 5 x − y = − 5 Equation 2: − x + 2 y = − 21
Use substitution method We can solve this system of equations using either the substitution or the elimination method. Let's use the substitution method. From Equation 1, we can express y in terms of x :
y = − 5 x + 5
Now, substitute this expression for y into Equation 2:
− x + 2 ( − 5 x + 5 ) = − 21
Solve for x Simplify and solve for x :
− x − 10 x + 10 = − 21
− 11 x = − 31
x = 11 31 = − 1
Solve for y Now that we have the value of x , substitute it back into the expression for y :
y = − 5 ( − 1 ) + 5
y = 5 + 5
y = 10
Final Answer So, the solution to the system of equations is x = − 1 and y = 10 . Therefore, the correct answer is:
( − 1 , 10 )
Examples
Systems of equations are incredibly useful in real life. Imagine you're trying to figure out the break-even point for a small business. You have one equation for your costs (fixed costs plus variable costs per item) and another equation for your revenue (price per item times the number of items sold). Solving this system tells you exactly how many items you need to sell to cover all your costs, which is crucial for making informed business decisions. This concept extends to many areas, from mixing chemical solutions to optimizing investment portfolios.
