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Chapter 2: Problem 61

Find the intercepts. Then graph by using the intercepts, if possible, and a third point as a check. $$ 5 y-x=5 $$

Short Answer

Expert verified
The intercepts are (0,1) and (-5,0). The third point is (1,1.2).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Substitute x=0 in the equation: \[ 5y - 0 = 5 \] So, \[ 5y = 5 \] Divide by 5: \[ y = 1 \]Thus, the y-intercept is (0, 1).
02

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Substitute y=0 in the equation: \[ 5(0) - x = 5 \]So, \[ -x = 5 \]Multiply by -1: \[ x = -5 \]Thus, the x-intercept is (-5, 0).
03

Find a third point

Choose another value for x or y to find a third point for accuracy in graphing. Let's use x=1. Substitute x=1 in the equation: \[ 5y - 1 = 5 \]So, \[ 5y = 6 \]Divide by 5: \[ y = \frac{6}{5} = 1.2 \]Thus, the third point is (1, 1.2).
04

Graph the equation

Plot the intercepted points (0,1) and (-5,0) on a graph. Use the third point (1,1.2) to check the accuracy of the line. Draw the straight line passing through these three points.

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Key Concepts

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

y-intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. This happens when the value of x is zero.
For the given equation, \(5y - x = 5\), we find the y-intercept by plugging x = 0 into the equation.
\[5y - 0 = 5\] which simplifies to \[5y = 5\]
Dividing both sides by 5 results in \[y = 1\]. So, the y-intercept is the point (0, 1).
This means our line crosses the y-axis at the point where y is 1.
x-intercept
The x-intercept of a linear equation is the point where the line crosses the x-axis. This happens when y is zero.
To find it for \(5y - x = 5\), set y to 0:
\[5(0) - x = 5\] simplifies to \[-x = 5\].
Multiplying both sides by -1 gives us \[x = -5\].
So, the x-intercept is the point (-5, 0). This means our line crosses the x-axis at the point where x is -5.
coordinate points
Coordinate points are essential in graphing as they mark specific locations on the graph. Each point is given in (x, y) format, which indicates its position along the x-axis and the y-axis.
For our equation \(5y - x = 5\), we already found two intercepts: (0, 1) and (-5, 0).
Additionally, finding a third point helps ensure our graph is accurate.
Using x = 1, we substitute into the equation:
\[5y - 1 = 5\],
which simplifies to \[5y = 6\] and then \[y = \frac{6}{5} = 1.2\].
This gives us the coordinate point (1, 1.2).
These three points, (0, 1), (-5, 0), and (1, 1.2), will help us accurately draw the line.
line graph
A line graph visually represents a linear equation by plotting points and connecting them with a straight line.
To graph \(5y - x = 5\), we start by plotting the intercepts: (0, 1) and (-5, 0).
Then, plot a third point, like (1, 1.2), for accuracy.
After plotting these points, draw a straight line through them. This line represents all possible solutions to the given equation.
Line graphs help us easily see the relationship between variables and understand how changes in x affect y.

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