Equations - True or False


Equations

Definition: An equation is a statement about two or more expressions stating that they are equal. Simply put, if there is an equal sign “=” then its an equation. It can be true like saying apple = apple or it can be false, like apple = sloth.

Question 1

Can you determine whether or not the equation, 3x - 10 = 6 - x, is true? Explain your reasoning.

We have 3x - 10 = 6 - x, or 3x - 10 is the same as 6 - x. Writing it in words often makes it clearer.

3 times (something we don’t know) minus 10 is the same as 6 minus (something we don’t know)

We do not know what x is, so there is no possible way for us to know if they are the same.

Therefore, we can’t determine whether or not it is true.


If x = 5, will the equation be true? Explain your reasoning.

Now we know a value of x, so we can work out now if the equation is true for the value of x,

x = 5

3(5) - 10 = 6 - (5)

As always, remember the order of operations!

15 - 10 = 6 - 5,

5 = 1,

As we know, 5 does not equal 1, so we can determine that the equation is not true for x = 5.


The equation is true for x = 4. Substitute x = 4 into the equation to show it is true.

The question is telling us that the equation is true when x = 4 in words,

3 times (4) minus 10 is the same as 6 minus (4),

So, sub in the value x = 4 into,

3x - 10 = 6 - x

3(4) - 10 = 6 - (4) =

12 - 10 = 6 - 4 =

2 = 2,

As we know, 2 does in fact equal 2, this means the equation is true for x = 6

Subbing In

When we are “subbing” in an x-value it can help to replace the x in the expression first with brackets, for example:

Sub x = 2 into the expression 3x + 1

So, first, write the expression as:

3( ) + 1

This can make it easier to see where we sub in our x value and how. So fill in the 2 from x = 2

3(2) + 1 =

6 + 1 = 7

Simply put, algebra uses basic mathematical knowledge, like 2 = 2, and changes it to work out variables we do not yet know. e.g. 2 = something + 1, we know that something must be 1. Now while this example is simple, it is representative of what happens even at the most advanced levels such as math that calculates radio frequencies or works out at what angle we need to launch a rocket ship at the reach the moon.

Solutions

Definition: The value of a variable is a solution to an equation if that value can be subbed into both sides of the equation to make that equation equal and true.

Question 2

Substitute the given value of the variable x into each equation given below. Determine whether or not that value of the variable is a solution to that equation.

2x + 4 = 10, x = 3

For these questions, we are given a value of x, and we check if it is a solution, meaning,

When we sub in the value of x into our equation, does it make both sides equal? If yes, it is a solution.

Order of operations is an easy place to make a mistake.

Sub x = 3 into 2x + 4 = 10

2(3) + 4 = 10

6 + 4 = 10

10 = 10,

So, x = 3 makes both sides of the equation equal, and therefore it is a solution to the equation.


{x - 10}/ 4 = - 6, x = 2

x= 2 into {x - 10}/ 4 = - 6

{2 - 10}/ 4 = - 6

Numerator first,

(-8)/ 4 = - 6

-2 = -6

x = 2 not a solution


3(x - 4) = 4(x + 1), x = - 16

x = - 16 into 3(x - 4) = 4(x + 1),

3((-16) - 4) = 4((-16) + 1)

Brackets first,

3(-20) = 4(-15)

-60 = - 60

x = -16 Is a solution


x2 + 1 = 3 - x, x = 1

x = 1 into x2 + 1 = 3 - x,

(1)^2 + 1 = 3 - (1)

Exponents first,

1 + 1 = 3 - 1

2 = 2

x = 1 is a solution


{2(x + 3)} / 5 - 1 = 6, x = 2

x = 2 into {2(x + 3)} / 5 - 1 = 6,

{2((2) + 3)} / 5 - 1 = 6

Numerator first,

{2(5)}/5 - 1 = 6

10/5 - 1 = 6

2 - 1 = 6

1 = 6

So, x = 2 is not a solution


(⅖)x + 2 = -(⅗) x + 7, x = 5

Subbing x = 5 into (⅖)x + 2 = -(⅗) x + 7

(⅖) (5) + 2 = -(⅗)(5) + 7

Multiplying the fractions out first,

(10)/5 + 2 = -(15)/5 + 7

2 + 2 = -3 + 7

4 = 4

x = 5 is a solution

Question 3

A student was checking to see if x = 8 was a solution to the equation 5x - 7 = 3x + 9, the student determined that x = 8 was not a solution to the equation, below is their working, determine whether or not they were right. If they were wrong identify which step they missed out.

x = 8 into 5x - 7 = 3x + 9,

5 x 8 - 7 = 3 x 8 + 9

5 x 1 = 3 x 17

5 = 51

Therefore x = 8 is not a solution

When questions ask to check an example of someone else's work to look for mistakes, it can often be easier to do the work yourself than it is to compare work and to find the mistake. However, still look through their work and see if you can spot it first although, in all likeliness, you will have to work it out yourself anyways!

So, working it out yourself,

x = 8 into 5x - 7 = 3x + 9,

5 x 8 - 7 = 3 x 8 + 9,

40 - 7 = 24 + 9

33 = 33

Therefore, x = 8 is a solution, so some of the students work must have been wrong, going through their work step by step,

x = 8 into 5x - 7 = 3x + 9,

5 x 8 - 7 = 3 x 8 + 9

Subbing in values was done correctly,

5 x 1 = 3 x 17

Now it looks like they did their subtraction and addition before their multiplication. We can see here how much difference mixing up the order of operations can make to our results!

Question 4

Is a = (-¼) a solution to the equation, -16a + 7 = -20a + 6

Note we are dealing with an “a” as the variable, that does not change our approach, and be careful subbing in a negative fraction

Subbing a = (-¼) into -16a + 7 = -20a + 6

-16(-¼) + 7 = -20(-¼) + 6

4 + 7 = 5 + 6

11 = 11

Therefore the equation is true for a = (-¼)