Question

solve the following system of equations graphically on the set of axes below.
y = -x + 4
y = 2x - 5
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze \( y = -x + 4 \)

This is a linear equation in slope - intercept form (\(y=mx + b\)), where the slope \(m=- 1\) and the y - intercept \(b = 4\). To graph this line, we can find two points. When \(x = 0\), \(y=-0 + 4=4\), so the point \((0,4)\) is on the line. When \(y = 0\), \(0=-x + 4\), which gives \(x = 4\), so the point \((4,0)\) is on the line.

Step2: Analyze \( y=2x - 5 \)

This is also in slope - intercept form with slope \(m = 2\) and y - intercept \(b=-5\). When \(x = 0\), \(y=2(0)-5=-5\), so the point \((0, - 5)\) is on the line. When \(y = 0\), \(0=2x-5\), then \(2x=5\) and \(x=\frac{5}{2}=2.5\), so the point \((2.5,0)\) is on the line.

Step3: Find the intersection

The solution to the system of linear equations is the point of intersection of the two lines. By graphing the two lines (either by plotting the points we found or using the slope - intercept form to draw the lines), we can see that the lines \(y=-x + 4\) and \(y = 2x-5\) intersect at the point where \(x = 3\) and \(y=1\) (we can also solve it algebraically: set \(-x + 4=2x-5\), then \(-x-2x=-5 - 4\), \(-3x=-9\), \(x = 3\), and substituting \(x = 3\) into \(y=-x + 4\) gives \(y=-3 + 4=1\)).

Answer:

The solution to the system of equations is \(x = 3\), \(y = 1\) (or the ordered pair \((3,1)\)).