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Chapter 1: Problem 23

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=5 x-6 $$

Short Answer

Expert verified
The x-intercept is \(x = \frac{6}{5}\) and the y-intercept is \(y = -6\).

Step by step solution

01

Find the x-intercept

Set y to 0 in the equation, which gives: \[0 = 5x - 6\]. Solving for x gives: \(x = \frac{6}{5}\)
02

Find the y-intercept

Set x to 0 in the equation, which gives: \(y = 5*0 - 6\). Solving for y gives \(y = -6\)

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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 graph is the point where the graph crosses the x-axis. This is the value of x when y equals zero. To find the x-intercept of a linear equation, simply set y to zero and solve for x.

For the equation given in the exercise, \( y = 5x - 6 \), we find the x-intercept by setting \( y = 0 \):
  • \[ 0 = 5x - 6 \]
  • Solving for \( x \), we add 6 to both sides: \[ 5x = 6 \]
  • Then, divide by 5: \[ x = \frac{6}{5} \]
Thus, the x-intercept is at the point \( \left( \frac{6}{5}, 0 \right) \).
Remember, every linear equation will have an x-intercept unless the line is horizontal.
y-intercept
Unlike the x-intercept, the y-intercept of a graph is where the line crosses the y-axis. This is the point where x is zero. To find the y-intercept, you set x to zero and solve for y.

For the equation \( y = 5x - 6 \), we determine the y-intercept by:
  • Setting \( x = 0 \): \[ y = 5(0) - 6 \]
  • This simplifies to \( y = -6 \)
The y-intercept is therefore \( (0, -6) \).
Graphically, this means the line of the equation crosses the y-axis at -6. Each linear equation with a non-zero slope will have a y-intercept.
linear equations
Linear equations are equations that graph as straight lines on a coordinate plane. They have constant slopes and can be written in different forms. The most common form is the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Key characteristics of linear equations:
  • They model a constant rate of change.
  • The graph is a straight line, which may be horizontal, vertical, or slanted.
  • The equation can be rearranged to highlight different aspects, like the slope or intercepts.
In this exercise, the slope-intercept form \( y = 5x - 6 \) shows a slope of 5 and a y-intercept of -6. Understanding these properties helps in graphing the line easily.
graphing equations
Graphing linear equations is a useful way to interpret or visualize mathematical relationships. It involves plotting points on a coordinate grid and drawing a line through those points. Using intercepts is a quick way to graph these equations.

Steps to graph the equation \( y = 5x - 6 \):
  • Start by plotting the y-intercept (0, -6).
  • Next, use the x-intercept \( \left( \frac{6}{5}, 0 \right) \).
  • Draw a straight line through these points.
Besides intercepts, you can also use the slope to find another point on the line. With a slope of 5, you rise 5 units up and run 1 unit to the right from any point on the line. This ensures accuracy and consistency in your graphing. Through practice, translating linear equations into their graphical form becomes intuitive.

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Most popular questions from this chapter

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