91Ó°ÊÓ

In algebra, the slope-intercept form is a way of writing the equation of a line. This form is particularly useful for graphing because it clearly shows the slope and y-intercept of the line. The general form is: \[ y = mx + b \]. Here, \( m \) represents the slope, and \( b \) represents the y-intercept. The slope \( m \) indicates the rate at which the y-coordinate of a point changes as the x-coordinate changes. A positive slope means the line rises to the right, while a negative slope means it falls to the right. The y-intercept \( b \) is the value of the y-coordinate where the line crosses the y-axis.

To find the y-intercept \( b \), we can use a given point on the line along with the slope. In our example, the point is \((2, 5)\) and the slope \( m \) is \(-\frac{3}{4}\). Begin by substituting these values into the slope-intercept form equation \( y = mx + b \). This looks like: \[ 5 = -\frac{3}{4}(2) + b \]. Simplify the equation: \[ 5 = -1.5 + b \]. Adding 1.5 to both sides, we get: \[ 6.5 = b \]. Hence, the y-intercept is 6.5. This tells us the line will cross the y-axis at the point \((0, 6.5)\).

To graph the line, you start with the y-intercept \((0, 6.5)\), which is the first point. After this, use the slope to find more points. Remember, slope \( m \) is the ratio of the change in y to the change in x, written as \( \frac{\text{slope}}{\text{run}} \). Our slope \( -\frac{3}{4} \) means for every 4 units you move to the right (run), move 3 units down (rise). From the y-intercept \((0, 6.5)\), move 4 units to the right to reach \(4, 6.5\). Then move 3 units down to reach \((4, 3.5)\). With these two points plotted \((0, 6.5)\) and \((4, 3.5)\), draw a straight line through them. This line represents the equation \( y = -\frac{3}{4}x + 6.5 \) and should extend infinitely in both directions.