Question

solve the following system of equations graphically on the set of axes below.
$y = 2x - 1$
$x + 2y = 8$
plot two lines by clicking the graph.
click a line to delete it.
answer attempt 3 out of 2
solution:

Explanation:

Step1: Graph \( y = 2x - 1 \)

The equation \( y = 2x - 1 \) is in slope - intercept form (\( y=mx + b \)) where the slope \( m = 2 \) and the y - intercept \( b=-1 \). To graph this line, we start by plotting the y - intercept at \( (0,-1) \). Then, using the slope (rise over run), from \( (0,-1) \), we rise 2 units (up) and run 1 unit (right) to get the next point \( (1,1) \), and we can continue this process to draw the line.

Step2: Rewrite \( x + 2y = 8 \) in slope - intercept form

We solve \( x + 2y = 8 \) for \( y \):
\[

$$\begin{align*} 2y&=-x + 8\\ y&=-\frac{1}{2}x+4 \end{align*}$$

\]
The slope of this line is \( m =-\frac{1}{2} \) and the y - intercept is \( b = 4 \). We plot the y - intercept at \( (0,4) \). Then, using the slope, from \( (0,4) \), we rise - 1 unit (down) and run 2 units (right) to get the next point \( (2,3) \), and we draw the line.

Step3: Find the intersection point

The solution to the system of equations is the point where the two lines intersect. By graphing the two lines \( y = 2x-1 \) and \( y=-\frac{1}{2}x + 4 \), we can see that they intersect at the point \( (2,3) \). We can also verify this algebraically by substituting \( y = 2x - 1 \) into \( x + 2y=8 \):
\[

$$\begin{align*} x+2(2x - 1)&=8\\ x + 4x-2&=8\\ 5x&=8 + 2\\ 5x&=10\\ x&=2 \end{align*}$$

\]
Then substitute \( x = 2 \) into \( y = 2x-1 \), we get \( y=2\times2 - 1=3 \).

Answer:

The solution to the system of equations is \( (2,3) \)