Question

solve the following system of equations graphically on the set of axes below:
( y = 2x - 5 )
( x + 2y = 10 )

plot two lines by clicking the graph.
click a line to delete it.

answer attempt 1 out of 5
solution:
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Explanation:

Step1: Analyze the first equation \( y = 2x - 5 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 2 \) and the y - intercept \( b=- 5 \). To find two points on this line:

• When \( x = 0 \), \( y=2(0)-5=-5 \), so we have the point \( (0,-5) \).
• When \( x = 2 \), \( y=2(2)-5 = 4 - 5=-1 \), so we have the point \( (2,-1) \).

Step2: Analyze the second equation \( x + 2y=10 \)

We can rewrite it in slope - intercept form (\( y=-\frac{1}{2}x + 5 \)) by solving for \( y \):
\[

$$\begin{align*} x+2y&=10\\ 2y&=-x + 10\\ y&=-\frac{1}{2}x+5 \end{align*}$$

\]
To find two points on this line:

• When \( x = 0 \), \( y = 5 \), so we have the point \( (0,5) \).
• When \( x = 10 \), \( y=-\frac{1}{2}(10)+5=-5 + 5 = 0 \), so we have the point \( (10,0) \).

Step3: Find the intersection point (graphically)

We can also solve the system of equations algebraically to find the intersection point. Substitute \( y = 2x-5 \) into \( x + 2y=10 \):
\[

$$\begin{align*} x+2(2x - 5)&=10\\ x + 4x-10&=10\\ 5x&=10 + 10\\ 5x&=20\\ x&=4 \end{align*}$$

\]
Now substitute \( x = 4 \) into \( y = 2x-5 \):
\( y=2(4)-5=8 - 5 = 3 \)

Answer:

The solution to the system of equations is \( (4,3) \) (the point of intersection of the two lines \( y = 2x-5 \) and \( x + 2y = 10 \)).