Problem 78 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (1,8) \text { and }(-4,-2) $$

Step-by-Step Solution

Answer

The slope-intercept form of the line that passes through the points (1,8) and (-4, -2) is \(y = -2x + 10\).

Step 1: Calculate the Slope

The slope (m) of a line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). For the points (1,8) and (-4, -2), the slope \(m\) is \((-2 - 8) / (-4 - 1) = 10/(-5) = -2\).

Step 2: Determine the Y-Intercept

To find the y-intercept (b), substitute the slope (m) and the coordinates of one of the points into the slope-intercept equation \(y = mx + b\), and solve for \(b\). Using the point (1,8) and the slope \(m = -2\), the y-intercept \(b\) becomes \(8 = -2*1 + b\), so \(b = 10\).

Step 3: Write the Equation of the Line

Now that we have calculated the slope \(m=-2\) and the y-intercept \(b=10\), the equation of the line in slope-intercept form will be: \(y = -2x + 10\).

Key Concepts

Understanding Linear Equations

A linear equation is an equation that represents a straight line when graphed on a coordinate plane. The most common form of a linear equation is the slope-intercept form, represented as \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept.
A linear equation can be understood through its graphical representation, which is a straight line with a constant rate of change.
This type of equation is powerful because it helps us understand relationships between two variables, predicting one variable if the other is known. The key features that define a linear equation include:

• Constant slope: This means the rate of change between any two points on the line is the same.
• Linear relationship: The plot of the equation is a straight line.
• Y-intercept: The point where the line crosses the y-axis.

Understanding how to write and interpret linear equations is vital since they are foundational in fields such as algebra, mathematics, and everyday problem-solving.

Slope Calculation

Calculating the slope of a line is one of the most crucial steps in writing a linear equation.
The slope indicates the steepness of the line and how much \( y \) changes for a unit change in \( x \). The formula for slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
When you have two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, substitute these points into the formula to find the slope. For example, using the points (1,8) and (-4,-2):

• Subtract the \( y \) values: \(-2 - 8 = -10\)
• Subtract the \( x \) values: \(-4 - 1 = -5\)
• Now divide these results: \(-10 / -5 = 2\)

The slope, therefore, is \(-2\). This slope tells us that for every step we move horizontally, the line moves two steps downward vertically. A negative slope means the line goes down as we move left to right.

Y-Intercept Determination

The y-intercept, denoted as \( b \) in the slope-intercept form \( y = mx + b \), is the point where the line crosses the y-axis.
This value can be found after determining the slope. Substitute the x and y values from a point on the line into the slope-intercept formula, along with the known slope, to solve for \( b \). For instance, using point (1,8) and slope \( m = -2 \):

• Replace the slope and point in the equation: \( 8 = -2 \times 1 + b \).
• Solve for \( b \): \( 8 = -2 + b \) then add 2 to both sides to get \( b = 10 \).

The y-intercept is \( 10 \), meaning the line intersects the y-axis at the point (0,10).
Understanding how to determine the y-intercept is critical for graphing and interpreting linear equations, as it provides a clear starting point for drawing the line.