Problem 26 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Slope \(\frac{2}{3} ; \quad y\) -intercept 4

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{2}{3}x + 4 \).

Step 1: Identify the slope-intercept form

The equation of a line in slope-intercept form is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.

Step 2: Substitute the slope

Substitute the given slope \( m = \frac{2}{3} \) into the equation. This gives us \( y = \frac{2}{3}x + b \).

Step 3: Substitute the y-intercept

Substitute the given \( y \)-intercept \( b = 4 \) into the equation. This results in \( y = \frac{2}{3}x + 4 \).

Key Concepts

Linear Equation

A **linear equation** is a type of equation that models a straight line on a graph. It is characterized by the presence of variables of only the first degree, meaning their highest exponent is one. This makes it very predictable and easy to manipulate mathematically. Every linear equation can be written in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
To simplify, we often use the slope-intercept form, \( y = mx + b \).

• In this form, \( m \) represents the slope of the line.
• \( b \) represents the y-intercept or the point where the line crosses the y-axis.

By using the slope-intercept format, it's easy to visually interpret the line's behavior on a graph and make predictions.

Slope

The **slope** of a line is a measure of its steepness and direction. In the slope-intercept form of a linear equation, \( y = mx + b \), the slope is represented by \( m \). It tells us how much the y-value (vertical change) changes for each unit increase in the x-value (horizontal change). Here’s how to understand it:- If the slope is positive, the line ascends from left to right.- If the slope is negative, the line descends from left to right.- A larger absolute value of the slope makes the line steeper.For example, a slope of \( \frac{2}{3} \), as given in the exercise, indicates that for every 3 units you move right on the x-axis, you will move up 2 units on the y-axis. It's a direct indicator of the line's rate of change and helps in predicting where the line will go when graphed.

Y-Intercept

The **y-intercept** is a vital concept when interpreting or constructing a graph from a linear equation. It is the point where the line crosses the y-axis, which means the x-value at this point is zero. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept.

• The y-intercept gives you a starting point to draw a line.
• It indicates the value of \( y \) when \( x = 0 \).

For instance, in this exercise, the y-intercept is 4, meaning the line will cross the y-axis at the point \( (0,4) \). This information simplifies plotting the line on a graph, as you have a concrete starting point without needing to solve an equation.