Problem 11 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (-4,3),(-1,-7) $$

Step-by-Step Solution

Answer

The equation of the line in the slope-intercept form that passes through the given points (-4,3) and (-1,-7) is \(y = -\frac{10}{3}x - \frac{31}{3}\).

Step 1: Calculate the slope

The formula for the slope, \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is: \(m=\frac{{y_2-y_1}}{{x_2-x_1}}\). So, the slope between the points (-4,3) and (-1,-7) is: \(m=\frac{{(-7)-(3)}}{{(-1)-(-4)}} = -\frac{10}{3}\).

Step 2: Substitute a point and the slope into the slope-intercept form and solve for \(b\)

Let's use the point (-4,3). So, substituting \(m = -\frac{10}{3}\), \(x = -4\) and \(y = 3\) into the equation \(y = mx + b\), we get: \(3 = -\frac{10}{3}(-4) + b\).\nThis simplifies to: \(3 = \frac{40}{3} + b\). So, \(b = 3 - \frac{40}{3} = -\frac{31}{3}\).

Step 3: Write the final equation

After substituting \(m = -\frac{10}{3}\) and \(b = -\frac{31}{3}\) into the slope-intercept form \(y = mx + b\), the resulting equation is: \(y = -\frac{10}{3}x - \frac{31}{3}\).

Key Concepts

Understanding the Equation of a Line

The equation of a line is a formula that describes all the points that make up a straight line on a coordinate plane. A commonly used form for these linear equations is the slope-intercept form, represented as \( y = mx + b \). Here, \( m \) is the slope of the line, showing the steepness or angle of the line. \( B \) is the y-intercept that indicates where the line crosses the y-axis.
Understanding these two components helps draw or interpret a graph accurately, as you can directly see how the graph is structured. By having these values, you don’t need to calculate each point on the line separately. Just plug in any \( x \) value to find \( y \), and you have a point on the line.
This simple form allows you to easily predict or trace a line on paper or a digital graph.

How to Calculate Slope

Calculating the slope of a line requires two points on that line. The formula to find the slope, \( m \), between two points \((x_1, y_1)\) and \((x_2, y_2)\) is quite straightforward:

• Subtract the y-values: \( y_2 - y_1 \)
• Subtract the x-values: \( x_2 - x_1 \)
• Divide the difference in y by the difference in x: \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \)

Using the example points \( (-4, 3) \) and \( (-1, -7) \), you substitute them into the formula: \( m = \frac{-7 - 3}{-1 - (-4)} = -\frac{10}{3} \).
This result, \( -\frac{10}{3} \), is negative, showing the line descends from left to right. A positive sequence would mean the line ascends similarly. A larger number, whether negative or positive, marks a steeper line while numbers closer to zero suggest a gentler slope.

Substituting a Point in the Equation

Once you have the slope, the next step is to use it to find an equation for the line.
Taking the slope-intercept form \( y = mx + b \), you substitute both a point on the line \((x_1, y_1)\) and the slope \(m\) into the equation to solve for \( b \).
For example, using the point \((-4, 3)\) with the slope \( m = -\frac{10}{3} \):

• Replace \( y \) with 3 and \( x \) with -4 in the formula
• \( 3 = -\frac{10}{3}(-4) + b \)
• Simplify: \( 3 = \frac{40}{3} + b \)
• Solve for \( b \): \( b = 3 - \frac{40}{3} = -\frac{31}{3} \)

This process discovers the y-intercept \( b \), forming a complete equation \( y = -\frac{10}{3}x - \frac{31}{3} \). Now, with both the slope and intercept known, the line can be easily plotted. Each substitution verified that the points and slope are accurately represented in the equation.