EEI - Solving Linear Equations Lesson

Solving Linear Equations

Solving equations is one of the most fundamental of all algebraic concepts. In fact, if you think about it, you've been doing algebra since you were in Kindergarten. Remember how your teacher would give you problems like this?

3+ blank = 5

The only difference is now we call that box a variable! See, you really have been doing this for a long time!

We are going to review how to solve linear equations and investigate some possibilities when doing so.

Remember, when solving an equation with one variable, the goal is to get the variable(s) on one side of the equation and the constant terms on the other side. To do this you will use the Properties of Equality for Addition, Subtraction, Multiplication, and Division.

 

x-4=-6 x=-2 Addition Property of Equality: The -4 and the +4 cancel out on the left

5n+3=13 -3. -3. Subtraction Property of Equality 5n=10 /5. /5. Division Property of Equality n=2

 **Remember, when you are solving equations, you can always substitute your answer back in and make sure both sides are, in fact, equal to each other.

4(x+5)=-4 4x+20=-4 Distributive Property of Equality 4x+20-20=-4-20 Subtraction Property of Equality (written horizontally) 4x=-24 /4. /4 Division Property of Equality x=-6

Watch this Khan Academy video for additional practice problems:

In all of these problems, we got an answer for x; one, unique answer. Does that always happen? Let's see...

3(3-x)+5x=2(x+2) 9-3x+5x=2x+4 Distributive Property of Equality 9+2x=2x+4 Combine Like terms 2x-2x+9=2x-2x+4 Subtraction Property of Equality 9-4

WHAT??? 9 is not equal to 4. This is a situation where no matter what we substitute in for x, we will always get an untruth. We say that this problem has no solutions. In other words, no value for x will ever make it true!

Now, let's look at a different scenario.

6-y=5-y+1 6-y+y=5-y+1+y Addition Property of Equality 6=6

This is true, 6 = 6, and our variable cancelled out. This means that ANY value that we use for x will yield a true, balanced equation. We say that this problem has infinitely many solutions, because all real numbers make it true.

In summation, there are three possibilities for the answer to a linear equation.

  1. One solution
  2. No solutions
  3. Infinitely many solutions

Here are some problems for you!

When we solve an equation, we go through the steps of isolating the variable:

five x plus two equals seven. five x equals five. x equals one

When you are asked to solve for a variable, you follow the same steps. Sometimes you may not be able to combine like terms, but that is OK!

Solve the equations for v

Let's try a few more together:

Here are some practice problems to check your understanding:

  1. Solve A  = bh for b.
  2. Solve d = rt for r.
  3. Solve P = 2l + 2w for w.
  4. Solve A = (1/2)bh for h.

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Mr. Jackson would like to take his biology students on a field trip to Ziggy's Zany Zoo. Student tickets cost $5 and adult tickets cost $8. Write an equation to represent the total cost, C, of the field trip tickets.

Let's discuss some important parts of the equation:

c equals five s plus eight a

If 30 students and 4 adults attend the field trip, what will the cost for tickets be?

C = 5(30) + 8(4) Replace s with 30 and replace a with 4.

C = 150 + 32 Simplify.

C = 182

It will cost $182.00.

Mr. Jackson was given a budget of $300, he must take 6 adult chaperones. How many students can attend the field trip?

300 = 5s + 8(6) Replace C with 300 and replace a with 6.

300 = 5s + 48 Solve for s.

252 = 5s

50.4 = s

Since Mr. Jackson cannot take 0.4 of a student, he'll have to take only 50 because he does not have enough money to take a whole other student.

Creating Linear Equations to Solve Problems

Let's go back to our introduction question: You pay $40 a month to get 5GB of data to use for your phone. Last month, you went over by 35 MB and you are charged $0.50 per MB. How much do you owe?

So we know that we owe $40 plus an additional $0.50 x 35 = $17.50 so a total of $57.50.

Could you write an equation for any amount of data used?

Let x = MB of data used over 5GB. So then you know you would owe $40 plus an additional $.50 times x. So our linear equation is C = $0.5x + 40.

Notice a few things about this equation:

It is linear because the degree of x is 1. We introduced a new variable, C, which represents the total cost of the bill. Now you can plug in any value for x to figure out your cost.

Use the equation above to determine the following: If your bill was $60 last month, how many extra MB did you use?

Let C = 60 since we know that was the total cost. The unknown, or question is how many MB did we use, which is what we defined x to be:

60 = 0.5x + 40

-40            -40

20 = 0.5x

20/0.5 = 0.5x/0.5

x = 10

So we know that you used 40 extra MB during that month.
Watch this video to try a few different examples:

Write and Solve Linear Equations Practice

 

Solving Linear Equations Practice

If you would like to practice more problems and check your work, click Here. Links to an external site. Make sure you check your answers by clicking here and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

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