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Chapter 15: Problem 39

Determine the coordinates of the \(y\)-intercept of each equation. Then graph the equation. \(y=2 x-5\)

Short Answer

Expert verified
The y-intercept is (0, -5).

Step by step solution

01

Identify the Y-intercept

To find the y-intercept of the equation, set \(x\) to 0 and solve for \(y\).
02

Substitute x with 0

In the equation \(y = 2x - 5\), substitute \(x = 0\): \[ y = 2(0) - 5 \] This simplifies to: \[ y = -5 \]
03

State the Y-intercept

The coordinates of the y-intercept are (0, -5).
04

Plot the y-intercept

On a graph, plot the point (0, -5).
05

Determine another point

To graph the line, find another point by choosing a value for \(x\). For example, let \(x = 2\): \[ y = 2(2) - 5 = 4 - 5 = -1 \] The coordinates are (2, -1).
06

Plot the second point

On the same graph, plot the point (2, -1).
07

Draw the line

Using a ruler, draw a straight line through the points (0, -5) and (2, -1). This line represents the equation \(y = 2x - 5\).

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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 occurs when the value of x is zero. To find the y-intercept for a given equation, substitute x with 0 and solve for y. For example, in the equation
\texpression (\(\foo\));
\ty = 2x - 5,
substituting x = 0 results in y = -5. Thus,
the coordinates of the y-intercept are (0, -5).
The ability to find the y-intercept is crucial for graphing and understanding the behavior of linear equations.
coordinates
Coordinates are used to specify the location of points on a graph. They are usually written in the form (x, y), where x represents the horizontal position and y represents the vertical position. In the original equation,
y = 2x - 5,
two points were identified: (0, -5) and (2, -1).
Knowing how to identify and plot these coordinates is essential for graphing linear equations.
When plotting a coordinate, first find the x-value on the x-axis, then find the y-value on the y-axis, and finally mark the point where these two values meet.
linear equation
A linear equation represents a straight line on a graph. It typically takes the form y = mx + b, where m is the slope and b is the y-intercept. In the example,
y = 2x - 5,
the slope (m) is 2, indicating the line rises 2 units for every 1 unit it moves to the right.
The y-intercept (b) is -5, where the line crosses the y-axis.
Understanding the components of a linear equation allows you to predict its graph, slopes, intercepts, and overall behavior.
plotting points
Plotting points on a graph involves locating each point's coordinates accurately. Start by plotting the y-intercept, as it gives one point directly from the equation.
Then, find another point by selecting a different value for x, substituting it into the equation, and solving for y. For instance, using
y = 2x - 5,
choosing x = 2 results in y = -1.
The coordinates (2, -1) is another point on the line.
After plotting these points (0, -5) and (2, -1), draw a straight line through them to represent the linear equation.
Ensuring accuracy when plotting is vital for a correct representation of the equation.

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