Problem 69 - Algebra 1: Concepts and Skills

Question

Find the slope and the y-intercept of the graph of the equation. Then graph the equation. $$ 3 x-y=-5 $$

Step-by-Step Solution

Answer

The slope of the equation is 3 and the y-intercept is 5. For the graph, start at the point (0,5) on the y-axis and move up 3 units for each unit moved to the right.

Step 1: Conversion to Slope-Intercept Form

Start by rearranging the given equation, \(3x - y = -5\), into the slope-intercept form \(y = mx + c\). Subtract \(3x\) from both sides to isolate \(y\) on one side of the equation. This results in the equation: \(y = 3x + 5\).

Step 2: Identify the Slope and Y-intercept

The slope \(m\) of the line is the coefficient of the \(x\) term, which in this line is 3. The y-intercept \(c\) is the constant term, which is 5.

Step 3: Plotting the Graph

Plot the graph of the equation using the slope and y-intercept identified. Start by marking the y-intercept (0, 5) on the y-axis. The slope 3 means that for every increase of 1 unit in \(x\), \(y\) increases by 3 units. So from the y-intercept, keep moving upward 3 units for every 1 unit you move to the right, and plot these points. Connect the dots to form the line.

Key Concepts

Slope

The slope is a fundamental concept when dealing with linear equations. In simple terms, the slope describes the steepness and direction of a line. Think of it as the line's level of incline or decline as you move along the graph. The slope is often denoted by the letter \( m \) and is defined as the ratio of the "rise" (change in \( y \)) over the "run" (change in \( x \)).

Here's how it works:

• If the slope is positive, the line ascends from the left to the right.
• If it is negative, the line descends from the left to the right.
• When the slope is zero, the line is perfectly horizontal.
• A line with an undefined slope is vertical.

In our equation, which is now written as \( y = 3x + 5 \), the slope \( m \) is 3. This indicates that for each unit increase in \( x \), \( y \) increases by 3 units. Understanding this helps us predict the line's direction on the graph, making it a critical concept when graphing linear equations.

Y-Intercept

The y-intercept is another essential component in understanding linear equations. It represents the point where the line crosses the y-axis.
This is the value of \( y \) when \( x = 0 \).
In the slope-intercept form \( y = mx + c \), the y-intercept is denoted by \( c \).
The y-intercept is particularly important because it provides a starting point for graphing the equation.In our example, the y-intercept is 5, from the equation \( y = 3x + 5 \). This means the line will cross the y-axis at the point (0, 5). To graph any line, you begin by plotting the y-intercept on the graph.
You can think of it as the anchor point of the line, setting the stage for plotting the rest of the points using the slope.

Graphing Linear Equations

Graphing linear equations involves using the slope and y-intercept to plot the line accurately on a graph. Once you understand the slope and y-intercept, graphing becomes a straightforward task.
Here’s a step-by-step guide to graph a linear equation:

• Start with the y-intercept: Plot the y-intercept on the y-axis. For our equation, it's at (0, 5).
• Use the slope: From the y-intercept, use the slope to determine your next points. For a slope of 3, every time you move 1 unit to the right, you move 3 units upwards.
• Draw the line: Once you have a couple of points, draw a straight line through these points extending across the graph.

These steps ensure that you can accurately depict the equation on a graph. With practice, you'll be able to visualize how changes in the equation affect the line's graph.