Problem 24 - Algebra 2

Question

Write an equation in slope-intercept form that satisfies each set of conditions. passes through \((5,-3)\) and \((-2,0)\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{3}{7}x - \frac{6}{7}\).

Step 1: Find the Slope

To write the equation in slope-intercept form \(y = mx + b\), we first need to calculate the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1) = (5, -3)\) and \((x_2, y_2) = (-2, 0)\). Substituting the values, we get: \(m = \frac{0 - (-3)}{-2 - 5} = \frac{3}{-7} = -\frac{3}{7}\). So, the slope \(m = -\frac{3}{7}\).

Step 2: Use the Point-Slope Form

We use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\), with the slope from Step 1 and one of the given points, say \((5, -3)\). Substituting \(m = -\frac{3}{7}\), \(x_1 = 5\), \(y_1 = -3\), we get: \(y - (-3) = -\frac{3}{7}(x - 5)\). This simplifies to \(y + 3 = -\frac{3}{7}(x - 5)\).

Step 3: Simplify to Slope-Intercept Form

Distribute the slope on the right-hand side: \(y + 3 = -\frac{3}{7}x + \frac{15}{7}\). Then, isolate \(y\) to convert this equation to slope-intercept form: \(y = -\frac{3}{7}x + \frac{15}{7} - 3\). Convert \(3\) into \pound fraction with the same denominator (\(\frac{21}{7}\)): \(y = -\frac{3}{7}x + \frac{15}{7} - \frac{21}{7}\). Finally, combine the fractions: \(y = -\frac{3}{7}x - \frac{6}{7}\).

Key Concepts

Finding the Slope

To create a strong foundation in solving linear equations, understanding how to find the slope is crucial. The slope of a line measures its steepness and direction. Mathematically, it is represented as \( m \) and calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \),

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.
This formula essentially tells us how much \( y \) changes between the two points for every change in \( x \). It's the "rise over run" concept that you may have heard in class. For instance, if the slope is positive, the line ascends as it moves from left to right. Conversely, a negative slope means the line descends.
In our example, using the points \((5, -3)\) and \((-2, 0)\), we found the slope to be \(-\frac{3}{7}\). This value confirms that the line decreases as we move along the x-axis.

Linear Equations

Linear equations are the foundation of Algebra. They represent straight lines when plotted on a graph and can take multiple forms, one of which is the slope-intercept form:

• \(y = mx + b\).

In this equation, \( m \) is the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
Understanding linear equations is greatly beneficial since it helps to solve real-world problems involving constant rates, such as speed or cost per item, by simplifying complex relationships into a manageable form.
The given exercise teaches us to derive such equations by finding the slope, using the point-slope form, and then simplifying it into the widely known slope-intercept form. This process unveils the straight path a line takes through specific points on a plane.

Point-Slope Form

The point-slope form is a powerful tool to write an equation of a line when we know its slope and a point on the line. The formula is:

• \(y - y_1 = m(x - x_1)\),

where \(m\) is the slope and \((x_1, y_1)\) is a known point.
This form is particularly useful because it allows us to create a linear equation directly from raw data without needing the y-intercept immediately. Let's consider the point \((5, -3)\) with a slope of \(-\frac{3}{7}\).
By substituting into the point-slope formula, we derive: \(y + 3 = -\frac{3}{7}(x - 5)\). Once we expand and rearrange, it can be converted to the slope-intercept form, making it easier to interpret and graph. This conversion is often the last step in solving such exercises, providing a complete overview of the line's behavior on the graph.