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Chapter 2: Problem 36

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 5 x+2 y=10 $$

Short Answer

Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 5). Plot these points and draw a line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x.Given equation: \[5x + 2y = 10\]Set \(y = 0\):\[5x + 2(0) = 10\]\[5x = 10\]\[x = \frac{10}{5}\]\[x = 2\]The x-intercept is at the point (2, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y.Given equation: \[5x + 2y = 10\]Set \(x = 0\):\[5(0) + 2y = 10\]\[2y = 10\]\[y = \frac{10}{2}\]\[y = 5\]The y-intercept is at the point (0, 5).
03

Plot the intercepts

Plot the x-intercept (2, 0) and the y-intercept (0, 5) on the coordinate plane.
04

Draw the line

Draw a straight line through the plotted intercepts (2, 0) and (0, 5). This line represents the graph of the given equation \(5x + 2y = 10\).

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Key Concepts

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

x-intercept
The x-intercept is the point at which a line crosses the x-axis of the coordinate plane. To find it, we set the value of y to zero and solve for x in the given equation. For example, to find the x-intercept in the equation \(5x + 2y = 10\), we substitute y with 0. This simplifies to: \[5x + 2(0) = 10\] \[5x = 10\] \[x = 2\] Thus, the x-intercept is at the point (2, 0). This tells us where the line meets the x-axis. Being able to find the x-intercept is crucial for graphing linear equations accurately.
y-intercept
The y-intercept is where the line crosses the y-axis. To find this, we set the value of x to zero and solve for y. For the equation \(5x + 2y = 10\), set x to 0: \[5(0) + 2y = 10\] \[2y = 10\] \[y = 5\] Hence, the y-intercept is at (0, 5). This position indicates a specific point where the line intersects the y-axis. By identifying the y-intercept, you gain a critical piece of information for plotting the line on a graph accurately and understanding the behavior of the equation.
graphing linear equations
Graphing linear equations involves plotting points and drawing a line through them. For example, when graphing the equation \(5x + 2y = 10\), we first find the intercepts. The x-intercept is (2, 0) and the y-intercept is (0, 5). To graph:
  • Plot the x-intercept (2, 0) on the coordinate plane.
  • Plot the y-intercept (0, 5) as well.
  • Draw a straight line passing through these two points.
This line represents all solutions to the equation. Remember, linear equations will always graph out as straight lines. Visualizing equations helps in understanding their behavior and relationships between x and y values.
coordinate plane
The coordinate plane is a two-dimensional surface where we graph points, lines, and curves. It consists of two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin (0,0) and divide the plane into four quadrants. Each point on the plane is defined by a pair of numbers (x, y) known as coordinates. For instance, the point (2, 0) indicates 2 units along the x-axis and 0 units along the y-axis. The point (0, 5) indicates 0 units along the x-axis and 5 units up the y-axis. Understanding the coordinate plane is fundamental for visualizing and solving problems involving graphs of equations.

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