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Chapter 5: Problem 34

Sketch the graph of the given equation. Label the intercepts. $$x+3=y-4$$

Short Answer

Expert verified
The intercepts are (0, 7) and (-7, 0).

Step by step solution

01

Rewrite the equation in slope-intercept form

Start by moving the terms to isolate y on one side of the equation. The given equation is: \( x + 3 = y - 4 \) To isolate y, add 4 to both sides: \( x + 3 + 4 = y \) Simplify the right side: \( y = x + 7 \)
02

Identify the slope and y-intercept

From the equation in slope-intercept form, \( y = mx + b \), identify the slope \( m \) and the y-intercept \( b \). The slope \( m \) is 1. The y-intercept \( b \) is 7. This means the line crosses the y-axis at (0,7).
03

Find the x-intercept

The x-intercept is the point where the graph crosses the x-axis (where \( y = 0 \)). Set y to 0 in the equation and solve for x: \( 0 = x + 7 \) Subtract 7 from both sides: \( x = -7 \) So, the x-intercept is (-7, 0).
04

Plot the intercepts and draw the line

Plot the y-intercept (0, 7) and the x-intercept (-7, 0) on a coordinate plane. Draw a straight line through these two points to complete the sketch of the graph.

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Key Concepts

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

slope-intercept form
To graph a linear equation, it's helpful to first rewrite it in the slope-intercept form. This form is expressed as: \( y = mx + b \) where:
  • m represents the slope of the line
  • b represents the y-intercept
Rewriting the equation makes it easier to identify key features like the slope and y-intercept.
  • The slope indicates how steep the line is and the direction it goes (upward or downward).
  • The y-intercept shows where the line crosses the y-axis.
In our example, \( x + 3 = y - 4 \), (Step 1) you can isolate y to get \( y = x + 7 \), which is in slope-intercept form.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This occurs where the value of x is 0. You can always spot the y-intercept easily in the slope-intercept form, as it is represented by the 'b' value. In the equation \( y = x + 7 \), the y-intercept is 7. To find the y-intercept manually in any equation (Step 2), you can set x to 0 and solve for y. In this case:
\( y = (0) + 7 = 7 \). Thus, the y-intercept is the point \( (0, 7) \). Mark this point on your graph and it gives you a good starting reference for your line.
x-intercept
The x-intercept is where the line crosses the x-axis. This point occurs when the value of y is 0. To find the x-intercept (Step 3), set y to 0 in the equation and solve for x. For the equation \( y = x + 7 \), set y to 0:
\( 0 = x + 7 \) Subtract 7 from both sides to solve for x:\( x = -7 \) Hence, the x-intercept is the point \( (-7, 0) \). Plot this point on your graph plot. Finally, draw a straight line through the y-intercept \( (0, 7) \) and the x-intercept \( (-7, 0) \) to complete the graph (Step 4).

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Most popular questions from this chapter

Write an equation of the line satisfying the given conditions. Vertical line passing through \((4,-3)\)

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