Solve the system of linear equations by graphing.
[tex]
\begin{array}{l}
y=-\frac{1}{2} x+3 \
y=\frac{1}{2} x-2
\end{array}
[/tex]
Answer (2)
Set the two equations equal to each other: − 2 1 x + 3 = 2 1 x − 2 .
Solve for x : x = 5 .
Substitute the value of x into one of the equations to solve for y : y = 2 1 .
The solution to the system of equations is ( 5 , 2 1 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = − 2 1 x + 3
y = 2 1 x − 2
Our goal is to find the solution to this system by graphing, which means finding the point where the two lines intersect. We are also given four possible solutions and we need to identify the correct one.
Setting the Equations Equal To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously. We can do this by setting the two equations equal to each other:
− 2 1 x + 3 = 2 1 x − 2
Solving for x Now, let's solve for x :
− f r a c 1 2 x − 2 1 x = − 2 − 3
− x = − 5
x = 5
Solving for y Now that we have the value of x , we can substitute it into either of the original equations to find the value of y . Let's use the second equation:
y = 2 1 ( 5 ) − 2
y = 2 5 − 2
y = 2 5 − 2 4
y = 2 1
Finding the Solution So, the solution to the system of equations is x = 5 and y = 2 1 . This corresponds to the point ( 5 , 2 1 ) .
Identifying the Correct Option Comparing our solution to the given options, we see that the correct solution is ( 5 , 2 1 ) .
Final Answer Therefore, the solution to the system of equations is ( 5 , 2 1 ) .
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 ingredients in a recipe, or modeling supply and demand in economics. In this case, solving the system of equations helps us find the point where two different linear relationships intersect, which can represent a balance or equilibrium between two variables. For instance, if the equations represented the cost and revenue of a product, the solution would be the point where the company breaks even.
To solve the system of equations y = − 2 1 x + 3 and y = 2 1 x − 2 by graphing, plot both lines and find the intersection point. The intersection occurs at ( 5 , 2 1 ) , which is the solution to the system. This solution can also be verified algebraically by solving the equations simultaneously.
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