Problem 58 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (-4,-2) and (-6,5)\(;\) use slope-intercept form. form.

Step-by-Step Solution

Answer

Equation: \( y = -\frac{7}{2}x - 9 \)

Step 1: Find the Slope

To find the slope \( m \) of the line that passes through the points \((-4,-2)\) and \((-6,5)\), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the given points into this formula:\[m = \frac{5 - (-2)}{-6 - (-4)} = \frac{5 + 2}{-6 + 4} = \frac{7}{-2} = -\frac{7}{2}\]So, the slope \( m \) is \(-\frac{7}{2}\).

Step 2: Use Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We already found the slope \( m = -\frac{7}{2} \). We now need to find \( b \).

Step 3: Calculate the Y-Intercept (b)

Substitute the slope \( m = -\frac{7}{2} \) and one of the points, say \((-4, -2)\), into the equation \( y = mx + b \) to find \( b \): \[-2 = -\frac{7}{2}(-4) + b \-2 = 14/2 + b \-2 = 7 + b \b = -2 - 7 \b = -9\]Therefore, the y-intercept \( b \) is \(-9\).

Step 4: Write the Final Equation

Substitute both the slope \( m = -\frac{7}{2} \) and the y-intercept \( b = -9 \) into the slope-intercept form to get the equation of the line:\[y = -\frac{7}{2}x - 9\]This is the equation of the line in slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a specific way to express the equation of a straight line. This form is simple and widely used due to its ability to quickly convey two key aspects of a line: the slope and the y-intercept. The general formula for the slope-intercept form is:

\( y = mx + b \).
* \( m \) represents the slope of the line. * \( b \) is the y-intercept, which indicates where the line crosses the y-axis.
Using this form makes it easy to graph a linear equation, as you directly see how steep the line is (the slope), and where it intersects the vertical axis of the graph. If you know just the slope and the y-intercept, you can readily write the equation in slope-intercept form and sketch the line on a graph.

Slope Calculation

Calculating the slope of a line helps determine how angled the line is. This concept is foundational in understanding how linear equations behave on coordinate planes. The slope is expressed as \( m \), and is calculated by determining the difference in the y-values divided by the difference in the x-values between two points on the line. The formula used to determine the slope \( m \) is:

\( m = \frac{y_2 - y_1}{x_2 - x_1} \).
* \( y_2 \) and \( y_1 \) are the y-values of the two points.* \( x_2 \) and \( x_1 \) are the x-values of the same points.
For example, to find the slope between the points \((-4, -2)\) and \((-6, 5)\), you'd compute:
\[ m = \frac{5 - (-2)}{-6 - (-4)} = \frac{7}{-2} = -\frac{7}{2} \].
This negative slope indicates that the line decreases as you move from left to right.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis on a graph. This point is represented by \( b \) in the slope-intercept form equation \( y = mx + b \). Understanding the y-intercept is important for quickly identifying where a line meets the vertical axis without needing to plot several points.

To find the y-intercept when you already know the slope \( m \), you can substitute a known point from the line into the equation and solve for \( b \). For instance, with a point \((-4, -2)\) and slope \( m = -\frac{7}{2} \), plug into:
\[-2 = -\frac{7}{2}(-4) + b \] which simplifies to \[ b = -9 \].
This means the line intercepts the y-axis at \( y = -9 \). Knowing \( b \), you can now fully write the line's equation as \( y = -\frac{7}{2}x - 9 \).