Problem 4 - Algebra A Combined Function

Question

Write an equation of each line with the given slope and containing the given point. Write the equation in the slope-intercept form \(y=m x+b .\) See Example \(1 .\) Slope \(-4 ;\) through (2,-4)

Step-by-Step Solution

Answer

The equation of the line is \( y = -4x + 4 \).

Step 1: Understand slope-intercept form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our task is to find the equation of a line given a slope and a specific point on the line.

Step 2: Substitute the slope

We are given the slope \( m = -4 \). Substitute this value into the slope-intercept form equation to get \( y = -4x + b \).

Step 3: Use the given point

The line passes through the point (2, -4). Substitute \( x = 2 \) and \( y = -4 \) into the equation \( y = -4x + b \) to find the value of \( b \). This gives us \( -4 = -4(2) + b \).

Step 4: Solve for b

Simplify the equation from Step 3: \( -4 = -8 + b \). Add 8 to both sides to solve for \( b \). This results in \( b = 4 \).

Step 5: Write the final equation

Now that we have \( b = 4 \), substitute \( b \) back into our equation \( y = -4x + b \). The final equation of the line is \( y = -4x + 4 \).

Key Concepts

Line Equation

Understanding the equation of a line is crucial in algebra. There are various forms, like point-slope form or standard form, but one of the most popular is the slope-intercept form. This is given by the formula \( y = mx + b \). The letter \( y \) stands for the output or dependent variable, while \( x \) represents the input or independent variable. In simple terms, it tells us the relationship between \( x \) and \( y \) through a straight line on a graph.

The slope-intercept form is particularly useful because it provides clear insights into two important features of a line: its slope and \( y \)-intercept. This form makes understanding how a line behaves and where it crosses the \( y \)-axis straightforward. If you know a line's slope and \( y \)-intercept, you can easily graph it or even draw it precisely on a chart.

When you're given a slope and a point, to write the equation, you plug the slope into the equation and use the point to solve for \( b \) (as seen in the example given). After finding \( b \), substitute it back to complete your line equation.

Slope

The slope of a line is a measure of its steepness or incline. It tells us how much \( y \) changes for every unit increase in \( x \). In the equation \( y = mx + b \), the slope is represented by \( m \). In simple words, if you have a higher slope value, the line is steeper. Conversely, a smaller slope means the line is flatter.

One crucial aspect of the slope is its sign:

• Positive slope: As \( x \) increases, \( y \) also increases, and the line rises from left to right.
• Negative slope: As \( x \) increases, \( y \) decreases, the line falls from left to right, like our given example where the line has a slope of \(-4\).
• Zero slope: The line is horizontal, meaning \( y \) doesn’t change as \( x \) increases.
• Undefined slope: Found in vertical lines where \( x \) remains constant regardless of \( y \).

The slope is calculated by "rise over run," which means how much you move up or down divided by how much you move sideways to get from one point to another on the line.

Y-Intercept

The \( y \)-intercept is a vital part of the slope-intercept equation of a line: \( y = mx + b \). The intercept is represented by \( b \), denoting the point where the line crosses the \( y \)-axis.

By knowing the \( y \)-intercept:

• You identify the starting point of the line on the graph.
• It shows the value of \( y \) when \( x = 0 \).
• Helps in quickly drawing the line on a graph.

In the problem provided, once we found out that \( b = 4 \), it told us the line crosses the \( y \)-axis at \( 4 \). This insight is essential when plotting the line, as it gives a precise point to begin sketching. The \( y \)-intercept helps in understanding the line's position in the \( xy \)-plane.