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Chapter 5: Problem 102

Find the slope and the \(y\) -intercept of the graph of \(2 x-3 y=6\)

Short Answer

Expert verified
The slope is \( \frac{2}{3} \) and the y-intercept is \( -2 \).

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by solving \( 2x - 3y = 6 \) for y. First, subtract \( 2x \) from both sides: \( -3y = -2x + 6 \).
02

Solve for y

Next, divide every term by -3 to isolate \( y \). This gives \( y = \frac{2}{3}x - 2 \). Now the equation is in slope-intercept form.
03

Identify the Slope and y-Intercept

Compare the equation \( y = \frac{2}{3}x - 2 \) to \( y = mx + b \). The slope \( m \) is \( \frac{2}{3} \), and the y-intercept \( b \) is \( -2 \).

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Key Concepts

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

linear equation
A linear equation represents a straight line when graphed on a coordinate plane. The general form of a linear equation in two variables is: \[ Ax + By = C \]where:\[ A, B, \text{ and } C \] are constants and \( x, y \) are variables. To better understand the characteristics of the line, we often convert the linear equation into the slope-intercept form. The process generally involves algebraic manipulation. Linear equations have a unique trait: for every value of \( x \), we can find a unique corresponding \( y \). This relationship gives us a straight line when plotted.
slope
The slope of a line measures how steep the line is. In the slope-intercept form of a linear equation:
\[ y = mx + b \]
the slope is represented by \( m \). It indicates how much \( y \) changes for a unit change in \( x \).
When the slope is positive, the line rises from left to right.
A negative slope means the line falls from left to right.
In the given problem, the equation was restructured to:
\[ y = \frac{2}{3} x - 2 \].
Here, the slope, \( m \), is \( \frac{2}{3} \). For every increase of 1 unit in \( x \), \( y \) increases by \( \frac{2}{3} \) units.
y-intercept
The y-intercept of a line is the point where the line crosses the \( y \)-axis. In the slope-intercept form:
\[ y = mx + b \]
, \( b \) represents the y-intercept.
This is the value of \( y \) when \( x \) is 0.
For the equation \( y = \frac{2}{3} x - 2 \), the y-intercept, \( b \), is -2. This means the line crosses the \( y \)-axis at (0, -2).
Knowing the y-intercept helps in drawing the graph of the linear equation, as it gives a starting point on the graph.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying equations to find desired variables or to put the equation into a recognizable form.
To convert the given equation \( 2x - 3y = 6 \) to the slope-intercept form \( y = mx + b \), specific steps were taken:
  • Subtracting \( 2x \) from both sides: \( -3y = -2x + 6 \)
  • Dividing each term by -3 to isolate \( y \):
    \( y = \frac{2}{3} x - 2 \).

Through these steps, the equation transforms into a more familiar format, making it easier to identify the slope and y-intercept. Algebraic manipulation is a powerful tool in solving and understanding linear equations.

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