Solve the following system of equations graphically:
[tex]
\begin{array}{c}
y=-\frac{1}{2} x+2 \\
x-y=4
\end{array}
[/tex]
Answer (1)
Rewrite the second equation in slope-intercept form: y = x − 4 .
Graph both equations: y = − 2 1 x + 2 and y = x − 4 .
Identify the point of intersection: (4, -0).
Verify the solution by substituting the values into both equations: The solution is ( 4 , − 0 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = − 2 1 x + 2 x − y = 4
Our goal is to solve this system graphically, which means we need to plot both lines on a coordinate plane and find their point of intersection. The coordinates of this point will give us the solution (x, y) to the system.
Rewriting the Equations First, let's rewrite the second equation in slope-intercept form (y = mx + b) to make it easier to graph. We have:
x − y = 4
Subtract x from both sides:
− y = − x + 4
Multiply both sides by -1:
y = x − 4
Now we have both equations in slope-intercept form:
y = − 2 1 x + 2 y = x − 4
Analyzing the Equations Now, let's analyze the first equation, y = − 2 1 x + 2 . The y-intercept is 2, so the line passes through the point (0, 2). The slope is - 2 1 , which means for every 2 units we move to the right, we move 1 unit down. So, another point on the line is (2, 1).
For the second equation, y = x − 4 , the y-intercept is -4, so the line passes through the point (0, -4). The slope is 1, which means for every 1 unit we move to the right, we move 1 unit up. So, another point on the line is (1, -3).
Finding the Intersection Point By graphing these two lines, we can find the point where they intersect. The intersection point appears to be at (4, -0. The x-coordinate is 4 and the y-coordinate is -0.
Verifying the Solution To verify the solution, we can substitute x = 4 and y = -0 into both equations:
For the first equation:
y = − 2 1 x + 2
− 0 = − 2 1 ( 4 ) + 2
− 0 = − 2 + 2
− 0 = − 0
For the second equation:
x − y = 4
4 − ( − 0 ) = 4
4 = 4
Since the point (4, -0) satisfies both equations, it is indeed the solution to the system.
Final Answer Therefore, the solution to the system of equations is:
( 4 , − 0 )
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 has fixed costs of $4000 and variable costs of $1 per unit, and they sell each unit for $3, the system of equations would be:
Total Costs: y = x + 4000 Total Revenue: y = 3 x
Solving this system would tell the company how many units they need to sell to break even (where total costs equal total revenue). Graphing these equations helps visualize the point at which the company starts making a profit.
