Solve the following system of equations graphically:
[tex]
\begin{array}{c}
y=-\frac{1}{3} x+3 \
3 x-y=7
\end{array}
[/tex]
Answer (2)
Rewrite the equations in slope-intercept form.
Graph both lines on the same coordinate plane.
Identify the point of intersection.
Verify the solution by substituting the coordinates into both original equations. The solution is x = 3 , y = 2 .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = − 3 1 x + 3
3 x − y = 7
We need to solve this system graphically, which means finding the point where the two lines intersect.
Rewriting the Equations First, let's rewrite the second equation in slope-intercept form (y = mx + b) to make it easier to graph:
3 x − y = 7
Subtract 3x from both sides:
− y = − 3 x + 7
Multiply both sides by -1:
y = 3 x − 7
Now we have both equations in slope-intercept form:
y = − 3 1 x + 3
y = 3 x − 7
Analyzing the Slopes and Intercepts The first equation, y = − 3 1 x + 3 , has a slope of − 3 1 and a y-intercept of 3. This means the line goes down 1 unit for every 3 units it moves to the right.
The second equation, y = 3 x − 7 , has a slope of 3 and a y-intercept of -7. This means the line goes up 3 units for every 1 unit it moves to the right.
Finding the Intersection Point Graphically By graphing these two lines, we can find their intersection point. The intersection point appears to be at (3, 2).
Verifying the Solution To verify the solution, we can substitute x = 3 into both equations and see if we get y = 2 in both cases.
For the first equation:
y = − 3 1 ( 3 ) + 3 = − 1 + 3 = 2
For the second equation:
y = 3 ( 3 ) − 7 = 9 − 7 = 2
Since both equations give y = 2 when x = 3, the solution is indeed (3, 2).
Stating the Solution Therefore, the solution to the system of equations is x = 3 and y = 2 .
Examples
Imagine you're planning a party and need to rent a venue. Venue A charges a $50 setup fee plus $10 per guest, while Venue B charges a $100 setup fee but only $5 per guest. Solving a system of equations like this helps you determine how many guests you need to invite for the total cost to be the same at both venues. This helps you make an informed decision based on your budget and guest list.
To solve the system of equations graphically, we rewrote the second equation into slope-intercept form, graphed both lines, and found their intersection point at (3, 2). We verified this solution by substituting x = 3 into both original equations, confirming that y = 2 for both. Therefore, the solution is (3, 2).
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