Problem 31 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ 3 x-4 y=7 $$

Step-by-Step Solution

Answer

Rearrange to slope-intercept form: \( y = \frac{3}{4}x - \frac{7}{4} \), then graph.

Step 1: Rearrange the Equation

To graph the linear equation, first rearrange it into the slope-intercept form, which is \( y = mx + b \). Starting with the equation \( 3x - 4y = 7 \), solve for \( y \): Subtract \( 3x \) from both sides:\[ -4y = -3x + 7 \]Now, divide everything by \(-4\) to solve for \( y \):\[ y = \frac{3}{4}x - \frac{7}{4} \].

Step 2: Identify the Slope and Y-Intercept

From the equation \( y = \frac{3}{4}x - \frac{7}{4} \), identify the slope \( m \) and the y-intercept \( b \). Here, the slope \( m = \frac{3}{4} \) and the y-intercept \( b = -\frac{7}{4} \).

Step 3: Plot the Y-Intercept

On the coordinate plane, plot the y-intercept. Since the y-intercept \( b = -\frac{7}{4} \), plot the point \( (0, -\frac{7}{4}) \).

Step 4: Use the Slope to Find Another Point

The slope \( \frac{3}{4} \) means for every increase of 3 in \( x \), \( y \) increases by 4. Starting at \( (0, -\frac{7}{4}) \), move right 4 units and up 3 units to find a second point \( (4, -1) \).

Step 5: Draw the Line

With the points \( (0, -\frac{7}{4}) \) and \( (4, -1) \) plotted, draw a straight line through these points to graph the equation \( 3x - 4y = 7 \).

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and connecting them with a straight line. A linear equation is one that can form a straight line when graphed. These equations typically take the form of \( y = mx + b \) or are easily rearranged into this format. By identifying the slope and y-intercept, we can accurately sketch the line that represents the equation.

When graphing:

• Start by rewriting the equation in a form suitable for plotting, typically the slope-intercept form.
• Next, plot the y-intercept on the y-axis.
• Use the slope to determine another point on the line.
• Finally, draw a line through these points, extending in both directions.

Graphing linear equations is a crucial skill for visualizing how different relationships work in algebra.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations and is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is popular because it clearly shows these two essential characteristics of the line.

To rewrite an equation in this form:

• Isolate \( y \) on one side of the equation using algebraic operations.
• The coefficients and constants will give you the slope and y-intercept directly.

This understanding makes graphing much simpler. It allows you to quickly determine the line's steepness and where it crosses the y-axis.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the equation \( y = mx + b \), the y-intercept is \( b \). This is a crucial point because it helps anchor the line on the graph.

To find the y-intercept:

• Set \( x = 0 \) in the equation and solve for \( y \). This will give you the y-coordinate of the intercept.
• On a graph, the y-intercept is represented by the point \( (0, b) \).

Understanding the y-intercept simplifies the process of graphing by providing a starting point to draw the line.

Coordinate Plane

A coordinate plane is composed of two intersecting perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants and are used to plot points identified by pairs of numbers (coordinates), \( (x, y) \).

When working with a coordinate plane:

• The point \( (0, 0) \) is known as the origin, where the axes intersect.
• The x-coordinate represents horizontal positioning, and the y-coordinate represents vertical positioning.
• Points are plotted using these coordinates, and a line is drawn through connected points to graph a linear equation.

Mastering the use of the coordinate plane is fundamental for graphing equations and understanding spatial relationships in algebra.