Problem 47 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(-5, y\) -intercept (0,7)

Step-by-Step Solution

Answer

The equation of the line is y = -5x + 7.

Step 1: Understanding the problem

We need to find the equation of a line given its slope and y-intercept. The provided slope is -5, and the y-intercept is the point (0, 7).

Step 2: Recalling the slope-intercept form

The slope-intercept form of a line's equation is given by y = mx + b where m d is the slope and b d is the y-intercept.

Step 3: Substitute the slope

Substitute the given slope (-5) for m d in the equation, giving y = -5x + b .

Step 4: Substitute the y-intercept

Substitute the given y-intercept (7) for b d in the equation. This gives y = -5x + 7 .

Step 5: Finalizing the equation

The equation of the line, incorporating both the slope and the y-intercept, is y = -5x + 7 .

Key Concepts

Linear Equations

Linear equations are like a simple set of instructions for a line on a graph. These equations describe a straight path and have a formula that can be easily followed. Every linear equation can be written in various forms, with the slope-intercept form being the most popular because of its simplicity. This version often gives quick insights into a line's angle and starting point on the graph.

A linear equation in slope-intercept form looks like this: \( y = mx + b \). Here, the letter \( y \) represents the value on the vertical axis, and \( x \) is the value on the horizontal axis. The letters \( m \) and \( b \) have special roles. They define the line's characteristics such as its tilt and where it crosses the vertical axis.

Linear equations are incredibly useful both in math and real-world scenarios. They allow us to predict and understand patterns and behaviors in a reliable way.

Slope

The slope of a line is a value that shows how steep the line is. With a linear equation in slope-intercept form, the slope is represented by the letter \( m \).

The slope tells us two things:

• How much the line rises or falls as it moves along the \( x \)-axis.
• Whether the line is moving upwards or downwards.

In our example, the slope is \(-5\), which means the line falls steeply as we move from left to right. The negative sign indicates a downward trend.

Understanding slope allows you to predict how one variable changes in relation to another. For example, in a business scenario, a negative slope might indicate a decrease in profit for every new product made. It's a powerful tool for interpreting data quickly.

Y-intercept

The y-intercept is a critical point where the line crosses the vertical \( y \)-axis. It's represented by the letter \( b \) in the slope-intercept form of a linear equation \( y = mx + b \).

In our specific equation, the y-intercept is \(7\). This tells us that at the start, when \( x = 0 \), the value of \( y \) is \(7\). Think of the y-intercept as the starting line for all potential scenarios of this equation.

This point is very helpful because:

• It gives a position reference on the graph.
• It helps in quickly sketching the line.

In practical situations, the y-intercept can represent an initial amount before any changes occur. For example, it can denote the starting amount in a financial context before any earnings or losses are considered.