Wize Grade 11 Mathematics Textbook > Review Material

Review of Linear Equations

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Review of Linear Equations
What is a Linear Equation?
A Linear Equation is any equation that can be written in the form y=ax+by=ax+by=ax+b where the values of aaa and bbb are real numbers.

When a linear equation is written in this way, we say it is in Slope-Intercept Form. 

Example 1

The following are all Linear Equations. .  
y=3x+2y=3x+2y=3x+2
y=23x−7y=\frac{2}{3}x-7y=32​x−7
2y=4x+62y=4x+62y=4x+6, this can be rearranged to become y=2x+3y=2x+3y=2x+3
3x−5y=−13x-5y=-13x−5y=−1, this can be rearranged to become y=35x+15y=\frac{3}{5}x+\frac{1}{5}y=53​x+51​

Wize Tip
It is often easier to tell if an equation is not linear by looking at the x variable.  If the x variable is raised to a power, or is in the denominator, the equation is not linear.    

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Graphs of Linear Equations
The graph of a Linear Equation has the basic shape of a completely straight line.  This line continues on in both directions infinitely.
 
   

Slope:
This characterizes the steepness of a line.
In slope-intercept form y=ax+by=ax+by=ax+b the slope is the number aaa associated with the xxx variable.  
y-intercept:
This is where the graph of the line cross the y-axis.
In the slope-intercept form y=ax+by=ax+by=ax+b, the slope is the number bbb.

Example 2

Identify the slope and the y-intercept in the following linear equations.  

1.  y=4x−5y=4x-5y=4x−5   

Slope: 4
y-intercept: 5

2.  y=−12x+3y=-\frac{1}{2}x+3y=−21​x+3

Slope: −12-\frac{1}{2}−21​
y-intercept: 3

3.  3y−6x=123y-6x=123y−6x=12
  First we can rearrange the equation to become y=2x+4y=2x+4y=2x+4.

Slope:  2
y-intercept: 4

Wize Tip
If a line has a positive value for the slope it will rise from the lower left to the upper right.  If a line has a negative value for the slope will fall from the upper left to the lower right.  

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Slope and y-intercept of Linear Equations
Calculating the slope using the graph
To calculate the slope of a line use the formula
slope=change in ychange in x\boxed{\text{slope}=\frac{\text{change in } y}{\text{change in }x}}slope=change in xchange in y​​
Pick two points on the line
Measure the change in the y and x direction.
Use + for up or right
Use - for down or left  
Use the formula to divide change in y over change in x

Wize Tip
Many people like to memorize this formula by saying that slope is the "rise" over the "run" like this 
slope=riserun\displaystyle \text{slope}=\frac{\text{rise}}{\text{run}}slope=runrise​

Example 1


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Calculating the y-intercept the graph
To calculate the y-intercept we can usually see where the graph of the linear equation crosses the y-axis.  Even though this is a point, only the y-value is used in describing the y-intercept.  

Example 2