91Ó°ÊÓ

Understanding the slope-intercept form is crucial when solving systems of equations graphically. The slope-intercept form is written as \ y = mx + b \. Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.

The slope \( m \) indicates how steep the line is. It is calculated as the rise over the run, or the change in y over the change in x. For example, a slope of \( \frac{1}{4} \) means that for every 4 units you move horizontally (right), you move 1 unit vertically (up).

The y-intercept \( b \) is the point where the line crosses the y-axis. This point tells us the value of y when x is 0. For instance, if \( b = 2 \), it means the line crosses the y-axis at (0, 2).

Both the slope and y-intercept are essential for graphing linear equations due to their roles in determining a line's position and slope.

Graphing linear equations involves plotting points on a coordinate plane based on the equation's slope and y-intercept.

Let's consider the equation \( y = \frac{1}{4}x + 2 \). The y-intercept is 2, so the line will cross the y-axis at (0, 2). From there, use the slope \( \frac{1}{4} \) to find another point. Starting at (0, 2), move 4 units to the right (positive direction on the x-axis) and 1 unit up (positive direction on the y-axis) to reach the point (4, 3). Now connect these points with a straight line.

Similarly, for the equation \( y = \frac{3}{2}x - 3 \). The y-intercept is -3, meaning it crosses the y-axis at (0, -3). From there, with a slope of \( \frac{3}{2} \), move 2 units to the right and 3 units up to get the point (2, 0). Connect these points to draw the line.

By plotting and drawing the lines of both equations, their intersection point can be identified visually on the graph.

The intersection point is where two lines cross on the graph. This point represents the solution to the system of equations because it satisfies both equations simultaneously.

In this context, to find the intersection point of \( y = \frac{1}{4}x + 2 \) and \( y = \frac{3}{2}x - 3 \), graph both lines on the same coordinate plane. The lines will intersect when both equations yield the same values of x and y.

From the graphs, it's possible to estimate the intersection point. However, precise values can often be determined more accurately by solving the equations algebraically. Substitute the coordinates of the estimated intersection point back into both original equations to confirm they work for both.

For our example, if we identify the intersection point as (4, 3), we verify it by substituting into both original equations:
In \( 4 y = x + 8\): \( 4 * 3 = 4 + 8 \), which is true.
In \( 3 x - 2 y = 6 \): \( 3 * 4 - 2 * 3 = 6 \), also true.

Thus, (4, 3) is indeed the solution to both equations.