Problem 122 - Introductory Algebra for College Students

Question

Write the slope-intercept form of the equation of the line passing through \((-3,-4)\) and \((1,0) .\) (Section 3.5 Example 2 )

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line passing through \((-3,-4)\) and \((1,0)\) is \(y = x - 1\).

Step 1: Find the Slope of the Line (m)

To find the slope of the line (m), use the formula \((y_2 - y_1)/(x_2 - x_1)\). Here \(x_1 = -3\), \(y_1 = -4\), \(x_2 = 1\), and \(y_2 = 0\). So the slope \(m = (0 - (-4))/(1 - (-3)) = 4/4 = 1\).

Step 2: Substitute one Coordinate and the Slope into the Equation

After having computed the slope (m = 1), substitute the one of the coordinates and the slope into the slope-intercept equation. For instance, using point (-3,-4) to substitute into \(y = mx + c\) to solve for c. \(y = mx + c\) becomes \(-4 = 1*(-3) + c\), yielding \(c = -1\).

Step 3: Write the Final Equation

After finding the slope (m) and the y-intercept (c), substitute these values into the slope-intercept equation to form the line's equation, which becomes \(y = 1x - 1\), or just \(y = x - 1\).

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It's defined as the ratio of the change in the y-coordinates to the change in the x-coordinates between two distinct points on the line. In simple terms, it tells us how much the y-value (up and down) changes as we move across the x-value (sideways). To find the slope, we use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]where

• \( (x_1, y_1) \) and \((x_2, y_2)\) are points on the line.
• \(m\) stands for the slope.

In our exercise, given the points \( (-3,-4) \) and \( (1,0) \),we plug them into the slope formula:\[m = \frac{0 - (-4)}{1 - (-3)} = \frac{4}{4} = 1\]This tells us that as we move one unit to the right along the x-axis, we also move one unit up along the y-axis. So, the line rises evenly, making the slope appear simple and intuitive.

Equation of a Line

Once we have the slope of a line, we can write its equation. The equation of a line in the slope-intercept form is expressed as:\[y = mx + c\]where

• \(y\) represents the dependent variable.
• \(x\) is the independent variable.
• \(m\) is the slope of the line.
• \(c\) is the y-intercept, or the value of \(y\) when \(x = 0\).

Given the slope \(m = 1\), we can use one of the points, say (-3, -4), to find the y-intercept \(c\). By substituting these values into the equation \(y = mx + c\), we get:\[-4 = 1 \times (-3) + c\]Solving the equation, we find \(c = -1\). Therefore, the equation of the line is:\[y = 1x - 1\]This simplifies to \(y = x - 1\). This equation represents a straight line with a slope of 1 and a y-intercept at -1.

Y-Intercept

The y-intercept is a key feature in the equation of a line. It is the point where the line crosses the y-axis. This occurs when the value of \(x\) is zero. In the context of the slope-intercept form, \(y = mx + c\), the y-intercept is the constant \(c\).In our problem, after calculating we found the y-intercept to be \(-1\). This means that if we have the line drawn on a graph, it will cross the y-axis at the point \( (0, -1) \).The y-intercept provides a starting point for the line before considering the slope. By locating this point on a graph, you can then use the slope to determine the direction and steepness of the line. Let's visualize this in a simple way: the y-intercept is where any journey on the line begins when starting from the y-axis. Understanding this can be incredibly useful for interpreting and graphing linear equations.