Question

Decide whether the graphs of the two equations are parallel lines. Explain your answer. $$ 3 y-4 x=3,3 y=-4 x+9 $$

Short answer

The given lines are not parallel because they do not have the same slope.

Step-by-step solution

Convert Equations to the form `y = mx + b`

The first step is to transform both given equations into the slope-intercept form, `y = mx + b`. Doing so gives: \n\nFor the first equation `3y - 4x = 3`, divide through by `3` to solve for `y` and find the slope: \[y = \frac{4}{3}x + 1\]. \n\nFor the second equation `3y = -4x + 9`, divide through by `3` to solve for `y` and find the slope: \[ y = -\frac{4}{3}x + 3\].

Compare the Slopes

Now, the comparison is easy because the equations are in slope-intercept form. The slopes are the coefficients of `x`. In the first equation, the slope is `4/3` and in the second equation, the slope is `-4/3`.

Determine if the Lines are Parallel

Two lines are parallel if and only if they have the same slope. Since the slopes of the two given lines are not the same, we can conclude that these lines are not parallel.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when it comes to analyzing linear equations. This form is written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) indicates the y-intercept, which is the point where the line crosses the y-axis.

Converting an equation into this form simplifies the process of graphing it because you can readily identify the direction in which the line moves (upward or downward, depending on the sign and value of the slope) and where it intersects the y-axis. Positive values of \( m \) indicate an upward slope, while negative values suggest a downward slope. If \( m \) is zero, the line is horizontal.

Example of Slope-Intercept Conversion

If we have the equation \( 3y - 4x = 3 \), we can convert it to slope-intercept form by isolating \( y \):

• Divide each term by 3 to get \( y = \frac{4}{3}x + 1 \).

This tells us that the slope (\( m \)) of the line is \( \frac{4}{3} \), and it crosses the y-axis at \( (0, 1) \).

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that enables students to find the values that satisfy the equation. The goal is to isolate the variable (usually \(y\) or \(x\)) on one side of the equation.

When faced with an equation like \(3y - 4x = 3\), the objective is to express \(y\) in terms of \(x\) to identify how changes in the value of \(x\) will affect \(y\). Here are the steps to solve such an equation:

• Rearrange the terms to bring variables on one side and constants on the other.
• Isolate \(y\) by dividing by the coefficient in front of \(y\), yielding the equation in the form of \(y = mx + b\).

This method paves the way for graphing the equations and comparing features such as slope and intercepts.

Consequences of Slope in Solutions

The slope tells us how 'steep' the line is; a larger absolute value of the slope indicates a steeper line. When two lines have the same slope, they could potentially run parallel to each other, never intersecting, but only if the y-intercepts are different. If they have the same y-intercept, then they are the same line.

Graphing Linear Equations

Graphing linear equations provides a visual representation of the solutions to an equation. Let's focus on how to graph equations once they are in slope-intercept form, \(y = mx + b\).

To graph a linear equation, start by plotting the y-intercept on the y-axis. Then, use the slope to determine the direction and steepness of the line. The slope indicates how much the value of \(y\) changes for a unit change in \(x\).

Plotting Steps

For \(y = \frac{4}{3}x + 1\), the following steps will help you graph it:

• Plot the y-intercept (0,1) on the graph.
• Identify the slope as \(\frac{4}{3}\), which means for every three units you move to the right (positive direction on the x-axis), you move up four units on the y-axis.
• Draw the line through these points, extending it infinitely in both directions.

When two lines on a graph are parallel, they have the same slope. This concept is essential for visual learners, as it ties the algebraic concept of slope to a graphical representation, helping students understand the relationships between multiple equations.