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Chapter 1: Problem 67

Find the \(x\) and \(y\)-intercepts of the line, and draw its graph. \(5 x+2 y-10=0\)

Short Answer

Expert verified
x-intercept: (2, 0); y-intercept: (0, 5).

Step by step solution

01

Find the x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the value of y is 0. Set y to 0 in the equation and solve for x:\[5x + 2(0) - 10 = 0\]Simplify the equation:\[5x - 10 = 0\]Solve for x:\[5x = 10 \x = 2\]Thus, the x-intercept is at the point \((2, 0)\).
02

Find the y-intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is 0. Set x to 0 in the equation and solve for y:\[5(0) + 2y - 10 = 0\]Simplify the equation:\[2y - 10 = 0\]Solve for y:\[2y = 10 \y = 5\]Thus, the y-intercept is at the point \((0, 5)\).
03

Plot the Intercepts on the Graph

Now that we have both intercepts, plot them on a coordinate plane. Mark the x-intercept \((2, 0)\) on the x-axis and the y-intercept \((0, 5)\) on the y-axis.
04

Draw the Line

Using a straight edge, draw a line through the points \((2, 0)\) and \((0, 5)\). This line represents the equation \(5x + 2y - 10 = 0\). Ensure that the line extends beyond both points to cover more of the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the x-Intercept
The x-intercept is where a line crosses the x-axis. This point is fascinating because at the x-intercept, the value of the y-coordinate is always zero. This is significant because it allows us to solve for the x-value directly from the equation by setting y to zero. For instance, using the line equation given, which is: \[5x + 2y - 10 = 0,\]you plug in 0 for y, simplifying to \[5x - 10 = 0.\]From here, solve for x to find the x-intercept: \[5x = 10\]by dividing both sides by 5, resulting in \[x = 2.\]Thus, the x-intercept for this line is at the point \((2, 0)\). This means the line will make contact with the x-axis at this point.
Finding the y-Intercept
The y-intercept is a bit different from the x-intercept. This is the point where the line intersects the y-axis, and here, the x-coordinate is always zero. We find the y-intercept by setting x to zero in the equation. Taking the same equation as before:\[5x + 2y - 10 = 0,\]set x to 0, resulting in \[2y - 10 = 0.\]You then solve for y:\[2y = 10,\]which simplifies to \[y = 5\]after dividing both sides by 2. Therefore, the y-intercept is at the point \((0, 5)\). This tells us that the line will touch the y-axis at \((0, 5)\). Understanding this helps in visualizing how the line behaves in a graph.
Graphing Linear Equations
Graphing linear equations is a wonderful way to see math in action! Once you have the x- and y-intercepts, you can graph the line. Begin by plotting these points on a coordinate grid. For the equation we are considering, the intercepts are \((2, 0)\) and \((0, 5)\). These points are your guides.
  • First, place a dot at \((2, 0)\) on the x-axis.
  • Next, place another dot at \((0, 5)\) on the y-axis.
With these dots plotted, use a ruler or a straightedge to draw a line through them extending the line beyond these points. This process represents the visual form of the equation \(5x + 2y - 10 = 0\). Be sure the line is straight and extends across the grid to give a full picture of how the line behaves. Through graphing, you transform an equation into a visual representation you can explore physically.

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