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Slope-intercept form is a way to describe linear equations in a simple way. It's written as \( y = mx + b \). Here, \( m \) is the slope of the line, which shows how steep the line is. The \( b \) is the y-intercept, which is where the line crosses the y-axis.
To turn an equation into slope-intercept form, solve for \( y \) in terms of \( x \). Let's look at an example based on the exercise. For the equation \( -x + y = -2 \), add \( x \) to both sides. This gives you \( y = x - 2 \), where the slope \( m \) is 1, and the y-intercept \( b \) is -2.
The second equation \( 2x + y = 10 \) is transformed by subtracting \( 2x \) from both sides. This results in \( y = -2x + 10 \). The slope here is -2. The y-intercept is 10. Being able to convert equations into slope-intercept form makes analyzing and graphing them easier.

Graphing linear equations is about putting these equations onto a graph to see what they look like visually. Start with plotting the y-intercept because it is the easiest point.
For the first equation \( y = x - 2 \), start at point \( (0, -2) \). This point is where the line crosses the y-axis. With the slope \( m = 1 \), for every step right, you move one step up. This gives you another point like \( (1, -1) \).
For the second equation \( y = -2x + 10 \), begin with the point \( (0, 10) \) for the y-intercept. With a slope of \(-2\), for each move right, go down by two. This creates a path, giving you another point like \( (1, 8) \).
All these points help in drawing each line on the graph, showing the path these equations take. It's like connecting the dots!

Solutions to linear systems involve finding any points where the equations' graphs meet or cross. In simpler terms, it's the values for \( x \) and \( y \) that satisfy both equations at the same time.
In the example, when graphing \( y = x - 2 \) and \( y = -2x + 10 \), look for the intersection point. This point is where both lines cross or meet on the graph.
To check if a point, like \((-4, -2)\), is a solution, plug it into both equations. It needs to work in both for it to be true. Substituting \(-4\) for \(x\) and \(-2\) for \(y\) into the first equation means checking if \(-2 = -4 - 2\). This equation isn't true, as left-hand side (LHS) does not equal the right-hand side (RHS). The point is not a solution.
Finding solutions is crucial for understanding how different lines or equations relate to each other.