$\begin{array}{c}
y=-\frac{1}{3} x+3 \\
3 x-y=7
\end{array}$
Plot two lines by clicking the graph. Click a line to delete it.
Answer (1)
Rewrite the second equation in slope-intercept form: y = 3 x − 7 .
Set the two equations equal to each other: − 3 1 x + 3 = 3 x − 7 .
Solve for x : x = 3 .
Substitute x = 3 into one of the equations to find y : y = 2 . The intersection point is ( 3 , 2 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = − 3 1 x + 3 3 x − y = 7
The task is to plot these two lines and find their intersection point.
Rewrite the second equation First, let's rewrite the second equation in slope-intercept form to make it easier to compare with the first equation. The second equation is 3 x − y = 7 . Solving for y , we get y = 3 x − 7 .
Identify slopes and intercepts Now we have two equations in slope-intercept form:
y = − 3 1 x + 3 y = 3 x − 7
The slope of the first line is − 3 1 and the y-intercept is 3. The slope of the second line is 3 and the y-intercept is -7. Since the slopes are different, the lines intersect at a single point.
Find the x-coordinate of the intersection To find the intersection point, we set the two equations equal to each other:
− 3 1 x + 3 = 3 x − 7
Now, we solve for x :
− 3 1 x − 3 x = − 7 − 3 − 3 10 x = − 10 x = 3
Find the y-coordinate of the intersection Substitute x = 3 into either equation to find the corresponding y -value. Using the first equation:
y = − 3 1 ( 3 ) + 3 = − 1 + 3 = 2
So, the intersection point is ( 3 , 2 ) .
Final Answer The two lines intersect at the point ( 3 , 2 ) . The first line has a y-intercept of 3 and a slope of − 3 1 . The second line has a y-intercept of -7 and a slope of 3.
Examples
Understanding systems of linear equations is crucial in various real-world applications. For instance, consider a scenario where you're comparing two different phone plans. Each plan has a fixed monthly fee and a per-minute charge. By setting up a system of linear equations, you can determine the number of minutes you need to use each month for the plans to cost the same. The intersection point of the two lines represents the usage at which both plans are equally priced, helping you make an informed decision based on your average monthly usage.
