Problem 51 - 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. $$ x-7 y=21 $$

Step-by-Step Solution

Answer

The x-intercept is at the point (21, 0) and the y-intercept is at the point (0, -3).

Step 1: Find the x-intercept

To find the x-intercept, set \(y\) to 0 in the equation and solve for \(x\). This gives an equation \(x - 7(0) = 21\), so \(x = 21\). Therefore, the x-intercept is at the point (21, 0).

Step 2: Find the y-intercept

To find the y-intercept, set \(x\) to 0 in the equation and solve for \(y\). This gives an equation \(0 - 7y = 21\), so \(y = -3\). Therefore, the y-intercept is at the point (0, -3).

Step 3: Graph and label the line

Plot the points (21, 0) and (0, -3) on a graph. These are the points where the line crosses the x and y-axes respectively. Draw the line that passes through these points. This line represents the equation. Be sure to label the points.

Key Concepts

Graphing Lines

Graphing a line is the practice of translating a linear equation into a visual representation on a coordinate plane. Linear equations typically have the form \(ax + by = c\), which describes a straight line. Graphing lines helps us to better understand the relationship between variables in the equation. There are several important steps you need to follow when graphing a line:

• Identify the equation and its coefficients.
• Find the intercepts, if necessary.
• Plot any special or easy-to-find points, such as intercepts, on the graph.
• Draw a straight line through these points.

The graph is complete once you have a line intersecting the two axes at the intercepts you've found. This visual tool can make it easier to predict outcomes or understand how changes in one variable might affect another.

x-intercept

The \(x\)-intercept of a line on a graph is the point where the line crosses the \(x\)-axis. At this point, the value of \(y\) is always zero because the line has not moved up or down, only horizontally. To find the \(x\)-intercept:

• Set \(y = 0\) in the equation.
• Solve for \(x\).

In the exercise, we have the equation \(x - 7y = 21\). When we set \(y = 0\), it simplifies to \(x = 21\). This means the line crosses the \(x\)-axis at the point \((21, 0)\). Graphically, this point is vital for understanding how the line behaves relative to the horizontal axis. Knowing the \(x\)-intercept helps to graph the entire line by providing a fixed point along the \(x\)-coordinates.

y-intercept

Similarly, the \(y\)-intercept is the point at which the line crosses the \(y\)-axis. Here, the value of \(x\) will always be zero because the line intersects where it moves vertically. To find the \(y\)-intercept:

• Set \(x = 0\) in the equation.
• Solve for \(y\).

In our example, setting \(x = 0\) in the equation \(x - 7y = 21\) results in \(-7y = 21\), giving \(y = -3\). Therefore, the \(y\)-intercept is the point \((0, -3)\). This intercept is crucial for the graph because it shows us exactly where the line meets the \(y\)-axis, conveying critical information about the line's behavior as it moves along the horizontal direction. Like the \(x\)-intercept, it anchors the line on one of the main axes, aiding in accurately drawing and interpreting the entire line.