Use the information in the right-hand frame in the following examples.
Example 1: Consider these two equations:
10x -35 = 5 and 4x +14 = 30
Does x = 4 satisfy both equations?
Yes.
Does that mean that these are equivalent equations?
Yes, if this is the only solution. Let's see if that is the case.
Start with 10x - 35 = 5. Add 35 to both sides.
10x - 35 + 35 = 5 + 35, or
10x = 40.
Divide both sides by 10.
, or
x = 4 is the only solution of the first equation.
To solve 4x + 14 = 30, what should we do first?
We should isolate the 4x by subtracting 14 from both sides.
Do it.
4x + 14 - 14 = 30 - 14, or
4x = 16
What should we do next?
Isolate x by dividing both sides by 4.
Do it.
, or
x = 4 is the only solution of the second equation.
What do we call a pair of equations with exactly the same solution set?
Equivalent equations.
Are these first degree equations?
Yes.
Example 2: Solve 5t - 3 = 2t + 6
This equation is more difficult than the previous ones because the variable appears in two places. How can we change the equation so that the variable appears only on the left side?
We can subtract 2t from both sides.
Do it.
5t - 3 - 2t = 2t + 6 - 2t, or
3t - 3 = 6
That is an improvement. What do we do next?
Isolate the 3t by adding 3 to each side.
Do it.
3t - 3 + 3 = 6 + 3, or
3t = 9
What is the last step?
Isolating "t" by dividing both sides by 3.
Do it.
, or
t = 3
How do we check our solution?
By substituting it into the original equation.
Do that.
Using x = 3 in 5t - 3 = 2t + 6, we get
5*3 - 3 ? 2*3 + 6.
Note that we use a question mark instead of an equal sign here because we don't know yet whether the two sides are equal.
Do the operations on each side and compare the results.
We get
5*3 - 3 ? 2*3 + 6, or
15 - 3 ? 6 + 6, or
12 = 12.
Now we can use the equal sign,
and have shown that x = 3 is the solution.
Example 3: Now let's try one with a fraction:
What is our first step when there are fractions?
We need to "clear the denominators".
How do we do that?
By multiplying both sides by the least common denominator (LCD).
What is the LCD here?
This is the easy case. The LCD is 3.
Multiply both sides by 3.
, or
x - 6 = 12
Solve for x.
We add 6 to both sides:
x - 6 + 6 = 12 + 6, or
x = 18
Is it helpful to substitute x = 18 into the original equation?
Yes - that's how we check our solution.
Do it.
Substituting x = 18, we get
, or
6 - 2 ? 4, or
4 = 4