Problem 24 - Algebra 1: Concepts and Skills

Question

Find the slope and y-intercept of the graph of the equation. $$3 x+4 y=16$$

Step-by-Step Solution

Answer

The slope of the graph is \(-\frac{3}{4}\) and the y-intercept is \(4\).

Step 1: Rewriting in Slope-Intercept form

First, solve \(3x + 4y = 16\) for \(y\) to get it in the form \(y = mx + b\). Subtract \(3x\) from both sides of the equation to get \(4y = -3x + 16\). Divide each term by \(4\) to find \(y\). This results in the equation \(y = -\frac{3}{4}x + 4\).

Step 2: Identifying the Slope and Y-Intercept

From the equation \(y = -\frac{3}{4}x + 4\), it's clear that the slope \(m\) is \(-\frac{3}{4}\) and the y-intercept \(b\) is \(4\).

Key Concepts

Slope-Intercept Form

In algebra, the slope-intercept form is a common and convenient way of representing a linear equation. It is written as \( y = mx + b \) , where \( m \) is the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
To convert a linear equation from another form, such as standard form \( Ax + By = C \) , to the slope-intercept form, we solve the equation for \b>\( y \) . Let's look at the textbook example: \( 3x + 4y = 16 \) .

• First, rearrange the terms to solve for \b>\( y \) .
• Subtract \b>\( 3x \) from both sides to isolate terms with \b>\( y \) on one side.
• Now, divide every term by the coefficient of \b>\( y \) , which is 4, to solve for \b>\( y \) .

This procedure brings us to the slope-intercept form, revealing the slope \( -\frac{3}{4} \) and the y-intercept \( 4 \) . Understanding this form is crucial for graphing and analyzing linear equations quickly.

Linear Equations

Linear equations form the foundation for much of algebra and represent straight lines on a graph. They can have one or more variables, but when graphed in two dimensions, they only need two variables, usually \b>\( x \) and \b>\( y \) . One property of these equations is that any solutions yield points that lie along a straight line when plotted on a coordinate plane.
In the example \b>\( 3x + 4y = 16 \) , we're dealing with a two-variable linear equation. The solutions to this equation correspond to points on the line it represents. To understand these points, we often express the equation in slope-intercept form to easily identify the slope, which indicates the direction and steepness of the line, and the y-intercept, where the line crosses the y-axis.
In other situations, we might encounter linear equations in different forms, such as point-slope form or standard form, but we can always convert them to slope-intercept form for easier interpretation.

Graphing Linear Equations

Graphing linear equations is a visual way of representing their solutions. When we graph the equation of a line, we typically start by plotting the y-intercept on the y-axis. In our example, the y-intercept is \( 4 \) , so we would put a point at \( (0, 4) \) on the graph.
After plotting the y-intercept, we use the slope to determine the direction and angle of the line. Remember, slope is rise over run. In the case of the slope \( -\frac{3}{4} \) , we move down three units (the rise) and right four units (the run) from the y-intercept to plot the next point.

Common Mistakes to Avoid

• Not reversing the sign when moving from one side of an equation to the other side.
• Forgetting to plot the y-intercept at the correct location on the y-axis.
• Misinterpreting the slope as run over rise, instead of the correct rise over run.

Once we have two points, we can draw a straight line through them, extending it across the graph. This line represents all the possible solutions for the equation. With practice, graphing becomes an effective tool for solving and understanding linear equations.