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Chapter 1: Problem 70

Sketch the graph of each linear equation. Be sure to find and show the \(x\) - and \(y\) -intercepts. $$5 x-2 y=10$$

Short Answer

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

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x.\( 5x - 2(0) = 10 \Rightarrow 5x = 10 \Rightarrow x = 2 \).So, the x-intercept is (2, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y.\( 5(0) - 2y = 10 \Rightarrow -2y = 10 \Rightarrow y = -5 \).So, the y-intercept is (0, -5).
03

Plot the intercepts

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

Draw the line

Draw a straight line through the points (2, 0) and (0, -5) to sketch the graph of the equation.

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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 of a linear equation is where the graph crosses the x-axis. At this point, the value of y is always 0 because it lies on the x-axis. To find the x-intercept of any equation, set y to 0 and solve for x. For example, in the equation \(5x - 2y = 10\), substitute y with 0 to get \(5x - 2(0) = 10\). Simplifying this gives us \(5x = 10\), and solving for x, we get \(x = 2\). Thus, the x-intercept is \(2, 0\).
y-intercept
The y-intercept of a linear equation is where the graph crosses the y-axis. At this point, the value of x is always 0 because it lies on the y-axis. To find the y-intercept, set x to 0 and solve for y. Using the same example \(5x - 2y = 10\), substitute x with 0: \(5(0) - 2y = 10\). This simplifies to \(-2y = 10\). Solving for y, we get \(y = -5\). Consequently, the y-intercept is \(0, -5\).
coordinate plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It has two axes: the horizontal x-axis and the vertical y-axis. Each point on the plane is identified by an \(x, y\) coordinate, which represents its location relative to the axes. When plotting a linear equation, the x-intercept and y-intercept are two key points that help determine the line's position and slope. For instance, for the points \(2,0\) and \(0,-5\) plotted on the coordinate plane, we draw the line passing through these points to graph the equation.
linear equations
Linear equations are equations of the first degree, meaning they have no exponents higher than 1. They graph as straight lines on the coordinate plane and can be written in various forms, like the slope-intercept form \(y = mx + b\) or the standard form \(Ax + By = C\). The graph of a linear equation is determined by finding key points such as the x-intercept and y-intercept and then drawing a line through these points. With the equation \(5x - 2y = 10\), we found the intercepts (2, 0) and (0, -5). Plotting these on the coordinate plane and drawing a line through them gives the graph of the equation.

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