1. Linear Equations
Block 43

2. Linear Equations
Linear equations are the simplest equations that you deal with. A linear equation is a mathematical expression that has an equal sign and linear expressions.
Vocabulary of Linear Equations:
Linear Equation:

3. Linear Equations
A variable is a number that you don't know, often represented by "x" or "y”. A linear expression is a mathematical statement that performs functions of addition, subtraction, multiplication, and division.
Vocabulary of Linear Equations:
Variable & Linear Expression

4. Steps to Solve Linear Equations
Linear equations are the simplest equations that you deal with.
Explain to the participants that they probably already solved linear equations and just didn't know it. Back in their early years they probably saw exercises such as ❏ + 3 = 8. Once you'd learned your addition facts well enough, you knew that you had to put a “5" in the box.
Solving equations works in much the same way, but now you have to figure out what goes into the x, instead of what goes into the box. However, now the equations can be much more complicated, and therefore the methods you'll use to solve the equations will be a bit more advanced.

5. Steps to Solve Linear Equations
Combine like terms.
Isolate terms that contain the variable you wish to solve for.
Isolate the variable you wish to solve for.
Substitute your answer into the original equation to check that it works.
To perform these steps you will need to use a number of mathematical properties of addition, subtraction, multiplication, and division. The use of these properties, both in combining like terms and isolating terms and variables, need to be reviewed.

6. Combine Like Terms
Like terms are terms that contain the same variable or group of variables raised to the same exponent, regardless of their numerical coefficient.
Example: 3z z = z
Keeping in mind that an equation is a mathematical statement that two expressions are equal, in this step focus on combining like terms for the two expressions contained in an equation. Use the example: 3z z = z. First combine like terms in each expression of this equation.

7. Combine Like Terms Example: 3z + 5 + 2z = 12 + 4z
5z -4z + 5 = z -4z
z + 5 = 12
The two expressions in this equation are 3z z and z  3z z = z
Step 1 & 2 combine the three terms that contain the variable z: 3z, 2z, and 4z.
Combine 3z, and 2z on the left side of the equation, then subtract 4z from both sides.
(3z +2z) + 5 = z z + 5 – 4z = z – 4z giving you z + 5 = 12
Notice we subtracted 4z from both sides rather than 5z. We do this because consolidating in this manner left z positive.
However, subtracting 5z from both sides would also be correct.

8. Isolate the terms with the variable
Example: 3z z = z
z = 12 -5
z = 7
The main idea in solving equations is to isolate the variable you want to solve for. This means getting terms containing that variable on one side of the equation, with all other variables and constants "moved" to the opposite side of the equation. This is done by using addition and its inverse property of subtraction.
Spend some time talking about this property:
The Addition Property of Equality and Its Inverse Property of Subtraction
If a = b, then a + c = b + c 
If a = b, then a – d = b – d
In other words, adding the same quantity to both sides of an equation produces an equivalent equation. Since subtraction is simply adding a negative number, this rule applies when subtracting the same quantity from both sides.
For the problem we started it is now time to isolate “z”.
Step 3: To isolate z, choose one term containing the variable z (usually the smaller one) to subtract from both sides.
Subtract 5 from both sides to isolate the z.
z + 5 – 5 = 12 – 5
The final result is z = 7

9. Check Solution Example: 3z + 5 + 2z = 12 + 4z z = 7
3(7) (7) = (7)
To be sure the answer is correct, check it by substituting the solution back to the original equation,
3z z = z
The left side becomes: 3 (7) (7)= = 40
The right side becomes: (7) = = 40
Notice that the right and left sides are equal, therefore we have the correct solution. =
40 = 40

10. Practice Solving Linear Equations
38 = z + 15

11. Practice Solving Linear Equations
x + 3 = 8x + 19

12. Practice Solving Linear Equations

13. Practice Solving Linear Equations

14. Graphing Linear Equations: T-Chart
Steps to solving a Linear Equation using a T-Chart.
Graph y = 2x + 1
Step 1: draw a chart that looks a bit like the letter "T"
Remind participants that graphing linear equations is pretty simple, but only if they work neatly. Being messy often makes extra work and they will frequently get the wrong answer.
There are several ways to graph a linear equation. Using a T-Chart is one way. There are several steps to solving a inear equation using a t-chart.

15. Graphing Linear Equations: T-Chart
Graph y = 2x + 1
Step 2: Label the columns.
x y ( or 2x+1)
The left column will contain the x-values, and the right column will contain the corresponding y-values.
The first column will be where you choose your input (x) values; the second column is where you find the resulting output (y) values. Together, these make a an ordered pair, (x, y).

16. Graphing Linear Equations: T-Chart
Graph y = 2x + 1
Step 3: Select values for x, solve for y
x y ( or 2x+1)
1 2(1) + 1 or 3
0 2(0) + 1 or 1
-1 2(-1) + 1 or- 1
Remind participants to select at least three value, to verify (when you're graphing) that you're getting a straight line. They should select some positive, some negative and zero. Usually 1, 0, 1 (and sometimes 2 & -2) are good and easy values to use.
("Linear" equations, the ones with just an x and a y, with no squared variables or square-rooted variables or any other fancy stuff, always graph as straight lines. That's where the name "linear" came from!)

17. Graphing Linear Equations: T-Chart
Graph y = 2x + 1
Step 4: Ordered pairs
x y ( or 2x+1) ordered pairs
1 2(1) + 1 or 3 (1,3)
0 2(0) + 1 or 1 (0,1)
-1 2(-1) + 1 or (-1,-1)
Now there are 4 or 5 ordered pairs needed for graphing.

18. Graphing Linear Equations: T-Chart
Graph y = 2x + 3
Step 5: Draw your coordinate axis
Now that there are ordered pairs, you can draw your axes. Remind participants to use a straight edge or ruler. This way, they will not have a messy axes and inconsistent scales on the axes. Also, remind them to not "fake it" with their graphs—accuracy is important.
Finally remind them to draw their axes large enough so the graph will be easily visible. On a standard-sized sheet of paper (8.5 by 11 inches), they canto fit two or three graphs on a page.
Remember that the arrows indicate the direction in which the values are increasing, then label with an appropriate scale.

19. Graphing Linear Equations: T-Chart
Graph y = 2x + 1
Graphing Linear Equations: T-Chart
Graph y = 2x + 3
Step 5: Draw your coordinate axis
ordered pairs
(1,3)
(0,1)
(-1,-1)
Now plot (draw) the ordered pairs from the T-chart. Then draw a line through the points.
Remember "linear" equations graph as a straight line.

20. Practice Graphing Linear Equations
y = 7 – 5x

21. Practice Graphing Linear Equations

22. Practice Graphing Linear Equations
y = 3

23. Practice Graphing Linear Equations
x = -2

24. Linear Equations in 2 Variables
The equation y = 2x – 1 produces a graph that is a straight line. This equation is one example of a general class of equations called linear equations in two variables. The two variables are usually x and y.
Linear equations are easy to recognize because they obey the following rules:
The variables (usually x and y) appear only to the first power
The variables may be multiplied only by real number constants
Any real number term may be added (or subtracted, of course)
Nothing else is permitted!

25. Linear Equations in 2 Variables
A specific straight line can be determined by specifying at least two distinct points that the line passes through or it can be determined by giving one point that it passes through and somehow describing how “tilted” the line is.
There are infinite number of straight lines that can be draw on a graph.
To describe a particular line we need to specify two distinct pieces of information concerning that line.

26. Slope Intercept
The one point is called the intercept or y-intercept. The “tilt” is called the slope.
Definitions
Slope is defined as being the “steepness” of a line. Mathematically it is the ratio of the change in y (vertical change) to the change in x (horizontal change).

27. Slope
The slope of a line is a measure of how “tilted” the line is. For example, a highway sign might say something like “6% grade ahead.”
Ask the participants what this means (other than that you hope your brakes work)?
What it means is that the ratio of your drop in altitude to your horizontal distance is 6%, or 6/100. In other words, if you move 100 feet forward, you will drop 6 feet; if you move 200 feet forward, you will drop 12 feet, and so on.

28. Using Slope-Intercept
The slope-intercept form is the most frequent way used to express an equation of a line. If an equation is in slope-intercept form, it is easy to graph. Slope-Intercept form is y=mx+b. m is slope and b is y-intercept
In general, the slope intercept form assumes the formula: y= =mx+b

29. Using Slope-Intercept
The slope-intercept form is the most frequent way used to express an equation of a line. If an equation is in slope-intercept form, it is easy to graph. Slope-Intercept form is y=mx+b. m is slope and b is y-intercept
In general, the slope intercept form assumes the formula: y= =mx+b