Question

1. solve by graphing: \\(\

$$\begin{cases} y = x - 2 \\\\ x + y = 10 \\end{cases}$$

\\)

Explanation:

Step1: Analyze the first equation

The first equation is \( y = x - 2 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m = 1 \) and the y - intercept \( b=-2 \). To graph this line, we can start by plotting the y - intercept at the point \( (0,-2) \). Then, using the slope (rise over run, since \( m = 1=\frac{1}{1} \)), we can find another point. From \( (0,-2) \), we move 1 unit up and 1 unit to the right to get the point \( (1,-1) \), and we can draw the line through these points.

Step2: Analyze the second equation

The second equation is \( x + y=10 \). We can rewrite it in slope - intercept form by solving for \( y \): \( y=-x + 10 \). Here, the slope \( m=-1 \) and the y - intercept \( b = 10 \). To graph this line, we plot the y - intercept at \( (0,10) \). Using the slope \( m=-1=\frac{- 1}{1} \), from \( (0,10) \), we move 1 unit down and 1 unit to the right to get the point \( (1,9) \), and we draw the line through these points.

Step3: Find the intersection point

The solution to the system of equations is the point where the two lines intersect. We can also solve the system algebraically to find the intersection point (which should match the graph). Substitute \( y=x - 2 \) into \( x + y=10 \):

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Then substitute \( x = 6 \) into \( y=x - 2 \), we get \( y=6-2=4 \). So the intersection point is \( (6,4) \). When we graph the two lines, they will intersect at the point \( (6,4) \).

Answer:

The solution to the system of equations is \( x = 6,y = 4 \) or the ordered pair \( (6,4) \)