Problem 38 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ x-y+4=0 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = x + 4\). The slope is 1 and the y-intercept is 4. The line graph should pass through the y-intercept at \(y = 4\) with a slope of 1.

Step 1: Rearrange to Slope-Intercept Form

First, the equation needs to be rearranged into slope-intercept form. This is achieved by isolating \(y\) on one side of the equation. From \(x - y + 4 = 0\), we can subtract \(x\) from both sides and subtract 4 from both sides to get \(-y = -x - 4\). Then, we can multiply or divide every term by -1 to transform the equation into \(y = x + 4\).

Step 2: Identify the Slope and Y-Intercept

The slope-intercept form is \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept. Comparing this with the equation \(y = x + 4\), it can be observed that \(m = 1\) and \(c = 4\), that means, the slope of the line is 1 and the y-intercept is 4.

Step 3: Plot the Graph

When graphing the equation, start by plotting the y-intercept(0, 4) on the graph. Since the slope is 1, which is a ratio of rise over run (meaning for every step to the right on the x-axis, go one step up on the y-axis), plot a second point by moving 1 unit to the right and 1 unit up from the y-intercept. Then, draw a line through these two points to graph the equation.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and are equations of straight lines. These equations can be expressed in various forms, but one of the most popular is the slope-intercept form, denoted as \(y = mx + c\).

Here, \(m\) represents the slope of the line, and \(c\) is the y-intercept, or the value of \(y\) when \(x = 0\). To solve or rearrange an equation into this form, you typically aim to make \(y\) the subject of the equation.

Linear equations simplify the process of graphing straight lines and analyzing their properties. They help students understand the relationship between variables in a systematic and visual way. With practice, interpreting linear equations becomes intuitive, unlocking further understanding of algebraic concepts.

Graphing Equations

Graphing equations is a useful skill that visually represents solutions to an equation. When you graph a linear equation, you showcase all the possible solutions of that equation on a coordinate plane.

To graph a linear equation effectively using the slope-intercept form, follow these steps:

• Start by identifying and plotting the y-intercept (0, c), where the line will intersect the y-axis.
• Next, utilize the slope to determine another point on the line. The slope \(m\), expressed as a fraction \(\frac{rise}{run}\), guides you on how to move from the y-intercept. For every unit you move right on the x-axis (run), you move \(m\) units up along the y-axis (rise).

Once you have two points on the graph, draw a line through them, extending it across your coordinate plane. This line is the graphical representation of the linear equation. With practice, graphing becomes a quick way to understand and visualize relationships between variables in equations.

Slope and Y-Intercept

The slope and y-intercept are key components of the slope-intercept form of a linear equation. Understanding them helps in interpreting and graphing linear equations.

• Slope (m): The slope symbolizes how steep a line is. It is the rate at which \(y\) changes with respect to \(x\). In the slope-intercept form \(y = mx + c\), \(m\) represents this value. A slope of 1, for example, implies that for every increase of 1 in \(x\), \(y\) increases by 1.
• Y-Intercept (c): The y-intercept is where the line crosses the y-axis. In the form \(y = mx + c\), \(c\) indicates this point. It shows the value of \(y\) when \(x = 0\). This point is crucial for starting your graphing process.

By mastering these concepts, students can better understand the behavior of linear equations and use them to predict other points along the line, simplifying the graphing process.