Problem 39 - Algebra 1: Concepts and Skills

Question

Find the \(x\) -intercepts and the \(y\) -intercepts of the line. Graph the equation. Label the points where the line crosses the axes. $$ y=x+3 $$

Step-by-Step Solution

Answer

The x-intercept is -3 and the y-intercept is 3. The graph is a straight line passing through the points (-3,0) and (0,3).

Step 1: Find the x-intercept

Set \(y\) in the equation equal to zero to find the x-intercept. Therefore: 0 = x + 3. By solving this equation for \(x\), we get \(x = -3\).

Step 2: Find the y-intercept

Set \(x\) in the equation equal to zero to find the y-intercept. Therefore: y = 0 + 3. By solving this equation for \(y\), we get \(y = 3\).

Step 3: Construct the graph

Plot the line using the x-intercept (-3,0) and the y-intercept (0,3). First, plot the x-intercept at the point (-3,0). Then, plot the y-intercept at the point (0,3). Connect these two points with a straight line.

Step 4: Label the line

Lastly, label the line where it crosses the x-axis at (-3,0) and the y-axis at (0,3).

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis. At this point, the y-coordinate is zero because you are on the x-axis.
To find the x-intercept of the equation, you set the y-value to zero and then solve for x. For the equation \(y = x + 3\), you would set it up as follows:

• Replace \(y\) with 0: \(0 = x + 3\)
• Solve the equation for \(x\): \(x = -3\)

This tells you that the line crosses the x-axis at the point \((-3, 0)\).
Remember, the x-intercept is a vital component in graphing because it gives one end of the line on the graph.

Exploring the y-intercept

The y-intercept of a line is where the line crosses the y-axis. Here, the x-coordinate is always zero because you are on the y-axis. To find it for a linear equation, set \(x\) to zero and solve for \(y\).
In the equation \(y = x + 3\), carry out the following steps:

• Set \(x\) to zero: \(y = 0 + 3\)
• This simplifies to \(y = 3\)

So, the line crosses the y-axis at the point \((0, 3)\).
The y-intercept is crucial because it helps form the line's base on the graph and provides a clear point of reference.

The basics of graphing equations

Graphing an equation, particularly a linear equation, involves plotting points and drawing a straight line that represents all solutions to the equation.
Take the linear equation \(y = x + 3\); since we've found the x-intercept to be \((-3, 0)\) and the y-intercept to be \((0, 3)\), graphing becomes straightforward.

• Plot those two intercept points on a coordinate plane.
• Place a point at \((-3, 0)\) on the x-axis.
• Place another point at \((0, 3)\) on the y-axis.
• Connect these points with a straight line.

This line should extend indefinitely in both directions unless specified otherwise. Graphing equations not only shows the behavior of the equation visually but also helps to confirm your calculated intercepts when they sit correctly on the respective axes.