Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$3 y=6 x$$

Step-by-Step Solution

Answer

The solution for \(y\), the equation is \(y = 2x\). The slope of the line is \(2\) and the y-intercept is \(0\).

Step 1: Solve the Equation for y

Divide both sides of the equation \(3y = 6x\) by 3 to isolate y on one side. Doing that, the equation becomes \(y = 2x\).

Step 2: Identify the Slope

The slope (m) is the coefficient of x in the equation \(y = mx + c\). Here, in the equation \(y = 2x\), the coefficient of x is 2, as there is no number visibly being added or subtracted. So, the slope of the line is 2.

Step 3: Identify the y-intercept

The y-intercept (c) is the constant term in the equation \(y = mx + c\), it's the point where the line crosses the y-axis. In our equation \(y = 2x\), there is no constant term, so we can assume it to be 0. That means the line crosses the y-axis at the origin (0,0). So, the y-intercept of the line is 0.

Key Concepts

Linear Equation

A linear equation is essentially an equation that forms a straight line when it is plotted on a graph. In algebra, a linear equation in two variables can be expressed in the form of \( y = mx + c \), where \( m \) represents the slope and \( c \) denotes the y-intercept. The equation captures a relationship between two quantities, with each input value of \( x \) corresponding to an output value of \( y \).

• It consists of variables and constants.
• The degree of the variables is one, making the graph a straight line.
• Linear equations help in understanding how two quantities change together.

The process of solving a linear equation involves isolating variables and simplifying terms, as was seen in the exercise when we transformed \( 3y = 6x \) into the slope-intercept form, \( y = 2x \). This form is particularly useful for visualizing the linear relationship on a graph, making it easier to identify both the slope and the y-intercept.

Slope

The slope of a linear equation tells us how steep the line is on a graph. It measures the rate of change or how much \( y \) increases or decreases as \( x \) increases by one unit. In the slope-intercept form of a linear equation, \( y = mx + c \), the slope is denoted by \( m \).

• A larger slope value means a steeper line.
• A positive slope indicates the line rises from left to right, while a negative slope shows it falls.
• A slope of zero denotes a flat, horizontal line.

In the exercise, once we reorganized the equation to \( y = 2x \), it became apparent that the line's slope is 2. This means, for every unit increase in \( x \), \( y \) increases by 2 units, creating a consistent upward tilt to the line.

y-intercept

The y-intercept is the point where the line crosses the y-axis in a graph. In the slope-intercept form \( y = mx + c \), the 'c' term is the y-intercept. It tells us the value of \( y \) when \( x \) is zero.

• It's an important feature for graphing and interpreting the linear equation.
• A positive y-intercept means the line crosses above the origin, while a negative one crosses below.
• If the y-intercept is zero, the line passes through the origin (0,0).

In our specific instance of the equation \( y = 2x \), there is no separate constant term, hence the y-intercept is 0. This tells you that the line goes through the origin, giving us a clear starting point on the graph.