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Graphing linear equations involves plotting points on a coordinate plane to visually represent the equation. Each linear equation forms a straight line when graphed. To start, you usually arrange the equation in the slope-intercept form, which makes it easier to identify key graph features.
Plotting is straightforward when you know the slope and the y-intercept. Using the slope-intercept form, \( y = mx + b \), allows you to quickly find these:

• The 'm' represents the slope. It shows how steep the line is.
• The 'b' represents the y-intercept, where the line crosses the y-axis.

After identifying these, you can plot the y-intercept on the graph and use the slope to find another point. Draw a line through these points to complete your graph. By doing this for two equations, you can see where they intersect, which is often the solution to a system of equations.

Slope-intercept form is a way to write the equation of a line so that you can easily determine its slope and y-intercept. The generic format is \( y = mx + b \).
In this format:

• 'y' is the dependent variable you solve for.
• 'm' is the slope of the line (rise over run).
• 'x' is the independent variable.
• 'b' is the y-intercept, where the line crosses the y-axis.

For example, if we start with the equation \( x - 3y = -6 \), we can rearrange it to the slope-intercept form:

• First, isolate 'y' by moving terms around: \( -3y = -x - 6 \).
• Next, divide every term by -3 to solve for 'y': \( y = \frac{1}{3}x + 2 \).

Now, it's easy to see the slope (\( \frac{1}{3} \)) and the y-intercept (2). Graph this by plotting the y-intercept at (0, 2) and using the slope to find another point (e.g., (3, 3)). Draw a line through these points.

The intersection of lines on a graph represents the point where the two lines meet. This point often signifies the solution to a system of linear equations.
To find this intersection:

• First, graph both equations on the same coordinate plane.
• Identify the point where the lines cross.
• This point gives the values for both 'x' and 'y' that satisfy each equation.

For instance, consider the equations \( y = \frac{1}{3}x + 2 \) and \( x = -3 \). The second equation represents a vertical line passing through \( x = -3 \). When you graph both, you'll see the intersection at (-3, 1).
Verify this by substituting \( x = -3 \) into the first equation:
\( -3 - 3y = -6 \rightarrow y = 1 \). This shows that (-3, 1) satisfies both equations, confirming the intersection point is correct and is the system’s solution.