Problem 48 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ y=5 x+15 $$

Step-by-Step Solution

Answer

The x-intercept is \( -3 \) and the y-intercept is \( 15 \). The points where the line crosses the axes are labeled as \( (-3,0) \) for the x-intercept and \( (0,15) \) for the y-intercept.

Step 1: Find the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \). So in the equation \( y=5x+15 \), if we set \( y \) to 0, we find \( 0=5x+15 \). Simplifying this, we get \( x=-3 \). So the x-intercept is \( -3 \).

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \). So in the equation \( y=5x+15 \), if we set \( x \) to 0, we find \( y=5*0+15 \). Simplifying this, we get \( y=15 \). So the y-intercept is \( 15 \).

Step 3: Graph the equation and label the points

Now, draw the line based on the equation \( y=5x+15 \). Plot the x-intercept at \( -3 \) on the x-axis and the y-intercept at \( 15 \) on the y-axis. Label the x-intercept as \( (-3,0) \) and the y-intercept as \( (0,15) \).

Key Concepts

x-intercept

The x-intercept is the point where the graph of an equation crosses the x-axis. This is a special point because at this spot, the value of y is always zero. To find the x-intercept of a linear equation, like in our example equation \(y = 5x + 15\), we substitute y with zero and solve the resulting equation for x.

• Substitute \(y = 0\) in \(0 = 5x + 15\)
• Subtract 15 from both sides which gives \(5x = -15\)
• Divide both sides by 5 to solve for x, resulting in \(x = -3\)

Therefore, the x-intercept is \(-3\), or as a point, \((-3, 0)\). Finding the x-intercept is a useful strategy in graphing because it provides valuable information about where the line intersects the x-axis. This serves as one of the pivotal points to accurately graph a linear equation.

y-intercept

When graphing a line on a coordinate plane, the y-intercept is the point where the line crosses the y-axis. At the y-intercept, the value of x is always zero. We can find the y-intercept by setting x to zero in the given equation. Let's take a closer look using our example equation, \(y = 5x + 15\).

• Substitute \(x = 0\) into the equation to get \(y = 5*0 + 15\)
• Simplify the equation, which gives \(y = 15\)

Thus, the y-intercept is \(15\), and as a coordinate point, it's \((0, 15)\). The y-intercept provides us with the exact location where the line intersects the y-axis. This can often be one of the first points used when sketching a graph as it clearly indicates the starting point of the line on the y-axis.

equation of a line

The equation of a line is a mathematical way to describe a straight line on a graph. For lines on a two-dimensional plane, like those encountered in algebra, we often use the slope-intercept form: \(y = mx + b\). In this form, \(m\) represents the slope of the line and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

Our example equation, \(y = 5x + 15\), fits this form perfectly.

• The slope, \(m\), is \(5\). This means the line rises by 5 units for every one unit it moves to the right.
• The y-intercept, \(b\), is \(15\). This is where the line meets the y-axis.

Understanding the equation of a line allows us not only to graph it but also to predict the behavior of the line. We find the direction and steepness of the line with the slope, and we locate its initial point of contact with the axes. Thus, by knowing any linear equation in this form, you can sketch or interpret its graph effectively.