Solve for x.
$\begin{array}{l}
-\frac{1}{2} y=\frac{1}{2} x+5 \\
y=2 x+2
\end{array}$
Answer (1)
Multiply the first equation by -2 to get y = − x − 10 .
Set the two equations equal to each other: − x − 10 = 2 x + 2 .
Solve for x : 3 x = − 12 , so x = − 4 .
Substitute x = − 4 into y = 2 x + 2 to find y = − 6 . The solution is x = − 4 , y = − 6 .
Explanation
Understanding the Problem We are given a system of two linear equations:
− 2 1 y = 2 1 x + 5 y = 2 x + 2
Our objective is to solve this system of linear equations to find the values of x and y .
Eliminating the Fraction First, let's multiply the first equation by − 2 to eliminate the fraction:
− 2 1 y = 2 1 x + 5 ( − 2 ) × − 2 1 y = ( − 2 ) × ( 2 1 x + 5 ) y = − x − 10
Now we have two equations:
y = − x − 10 y = 2 x + 2
Solving for x Since both equations are solved for y , we can set them equal to each other:
− x − 10 = 2 x + 2
Now, let's solve for x . Add x to both sides:
− 10 = 3 x + 2
Subtract 2 from both sides:
− 12 = 3 x
Divide by 3:
x = − 4
Solving for y Now that we have the value of x , we can substitute it into either equation to solve for y . Let's use the second equation:
y = 2 x + 2 y = 2 ( − 4 ) + 2 y = − 8 + 2 y = − 6
So, y = − 6 .
Final Answer Therefore, the solution to the system of equations is x = − 4 and y = − 6 .
x = − 4 , y = − 6
Examples
Systems of linear equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of products to maximize profit, or modeling supply and demand in economics. For example, a company might use a system of equations to determine the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve these systems is crucial for making informed decisions in many fields.
