/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Graph each linear equation. $$... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 44

Graph each linear equation. $$ 6 x-5 y=18 $$

Short Answer

Expert verified
Plot the points \( (0, -\frac{18}{5}) \) and \( (5, \frac{12}{5}) \), then draw a line through them.

Step by step solution

01

- Write the equation in slope-intercept form

Start with the given equation: \[ 6x - 5y = 18 \] To convert this into slope-intercept form \( y = mx + b \), solve for \( y \). First, subtract \( 6x \) from both sides: \[ -5y = -6x + 18 \] Next, divide every term by -5: \[ y = \frac{6}{5}x - \frac{18}{5} \]
02

- Identify the slope and y-intercept

From the slope-intercept form \( y = mx + b \), the coefficient of \( x \) is the slope \( m \), and the constant term is the y-intercept \( b \). Here, the slope \( m \) is \( \frac{6}{5} \) and the y-intercept \( b \) is \( -\frac{18}{5} \).
03

- Plot the y-intercept

The y-intercept is the point where the line crosses the y-axis. Plot the point \( (0, -\frac{18}{5}) \) on the graph.
04

- Use the slope to find another point

From the y-intercept \( (0, -\frac{18}{5}) \), use the slope \( \frac{6}{5} \) to find another point. The slope \( \frac{6}{5} \) means that for every increase of 5 units in the x-direction, the y-value increases by 6 units. Starting from \( (0, -\frac{18}{5}) \), move 5 units to the right (x = 5) and 6 units up (\( y = -\frac{18}{5} + 6 \)). Thus, the new point is \( (5, -\frac{18}{5} + \frac{30}{5}) = (5, \frac{12}{5}) \). Plot this point on the graph as well.
05

- Draw the line

Draw a straight line through the two points \( (0, -\frac{18}{5}) \) and \( (5, \frac{12}{5}) \). This is the graph of the linear equation \( 6x - 5y = 18 \).

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Key Concepts

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

slope-intercept form
The slope-intercept form of a linear equation is one of the most common ways to represent straight lines. You'll see it written as: \( y = mx + b \). This format makes it easy to identify the 'slope' of the line and its 'y-intercept'.
Here's how to understand it:
  • y is the variable representing the dependent variable (usually how far up or down you are on the graph)
  • m is the 'slope' of the line. It shows the steepness or incline of the line.
  • x is the independent variable (usually how far left or right you are on the graph)
  • b is the 'y-intercept'. It's the point where the line crosses the y-axis.

When given an equation like \(6x - 5y = 18\), converting it to slope-intercept form makes graphing a lot simpler. We isolate y to get \[ y = \frac{6}{5}x - \frac{18}{5} \]. Now you can easily identify the slope (\(\frac{6}{5}\)) and y-intercept (\(-\frac{18}{5}\)).
slope
The 'slope' (m) of a line is a measure of its steepness. It's calculated as the 'rise' over the 'run', or how much y changes for a change in x. If the rise is 6 and the run is 5, the slope is \(\frac{6}{5}\).
Key things to remember about slopes:
  • Positive slope: The line goes up as you move from left to right.
  • Negative slope: The line goes down as you move from left to right.
  • Zero slope: The line is flat, meaning it鈥檚 horizontal.
  • Undefined slope: The line is vertical.

In our converted equation \( y = \frac{6}{5}x - \frac{18}{5} \), the slope \(\frac{6}{5}\) tells us that for every 5 units we move to the right (x-direction), we go up 6 units (y-direction). This helps us plot additional points.
y-intercept
The 'y-intercept' (b) is where the line crosses the y-axis. This occurs when x is 0. In our example, the y-intercept can be found directly from the slope-intercept equation: \( y = \frac{6}{5}x - \frac{18}{5} \).
The constant term \(-\frac{18}{5}\) is the y-intercept.
Key points to remember about the y-intercept:
  • It represents the value of y when x is 0.
  • It's crucial for plotting the initial point on the graph.

For the equation \( y = \frac{6}{5}x - \frac{18}{5} \), the y-intercept is \( -\frac{18}{5} \), or approximately -3.6. So, the first point to plot on the graph is \( (0, -\frac{18}{5}) \). From this point, you can use the slope to find other points and draw the line.

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