In this lesson we will become familiar and comfortable with the use of constants represented by letters. We will solve linear equations with these constants as if we knew the value of them. The unspecified constants are different to variables. Variables vary and constants remain constant! All these constants are doing is representing a singular value.

The solutions in the above animation are repeated with a more detailed explanation below.

Solve 4x + 7 = 27 for *x*.

Hint

Subtract both sides by 7

4x + 7 - 7 = 27 - 7

4x = 20

Divide both sides by 4

4x ÷ 4 = 20 ÷ 4

x = 5

Solve ax + b = c for *x*, where *a*, *b* and *c* are constants.

This may seem daunting! How can you **solve** something, with no numbers! We must remind ourselves what the symbols and letters in the equation mean that mathematics is a language. We are told *a*, *b* and *c* are constants, so this means they could be something like 1 or 1028 or -4/3, but we just aren't told specifically what they are.

Begin to get comfortable with having these constants representing constant values.

It can help by comparing ax + b = c to the equation in 1a) 4x + 7 = 27,

4x + 7 = 27

ax + b = c

Now, if we imagine *a*, *b* and *c* representing the similar values in 4x + 7 = 27

As we move on to solve ax + b = c, think how we would operate if we knew what the constant values were, just as we did in a)

ax + b = c

First, we would subtract the b from both sides

ax + b - b = c - b

ax = c - b

If it helps, think of c - b as a number as well, as we know, *c* and *b* are both numbers, so c - b would also be a number.

Divide both sides *a*,

ax

a

=

c - b

a

x =

c - b

a

There we have our solution, despite it maybe not looking like one. Remember, *a*, *b* and *c* are numbers, (c - b) ÷ a would be a number as well.

**Something extra**: as we were “pretending” this was the same as question 1a), let’s try subbing in our a the same as 4, *b* the same as 7 and *c* the same as 27 into (c - b) ÷ a

(27 - 7) ÷ 4 =

20 ÷ 4

5

Cool right!?!?

The solutions in the above animation are repeated with a more detailed explanation below.

Solve

x - a

b

+ c = d

for x with the constants *a, b, c*, and *d*.

Get into the mindset that as soon as you see “with the constants *a*....” immediately think of them as representing numbers. In the same way you might see *x* and start thinking about how you are going to solve it, see a constant represented by a letter and think that's a number I'll be working with, even though it's a letter!

If we are confused it might help to go over our inverse operations again,

So,

In order of what we would do to *x*,

- Subtract
*a* - Divide by
*b* - Add
*c*

So Inverse Operations, we do the opposite,

- Subtract
*c* - Multiply by
*b* - Add
*a*

So, for

x - a

b

+ c = d

Subtract *c* from both sides,

x - a

b

= d - c

Multiply by *b*,

x - a

b

b = (d - c)b

x - a = (d - c) b

Add *a*,

x - a + a = (d - c) b + a

x = (d - c) b + a

For the next question, solve the equation for x. However, **do not** carry out any of the operations when solving. Leave all the **numbers** in the original equation as is. For example, if you had x + 1 = 3, leave your answer as x = 3 - 1 not x = 2. You can use the distributive property.

Doing this, we should still be able to see the numbers we have in the equation in our answer.

The solutions in the above animation are repeated with a more detailed explanation below.

Solve ax + b = cx + d , for *x* with *a, b, c*, and *d* being constants.

We will use the same idea as we had in the example above with these constants. Think of the constants as numbers, just as we did in the previous question.

First, we would “get our *x* on one side”, so subtract cx from both sides.

(ax + b) - cx = (cx + d) - cx

ax - cx + b = d

Think of the a and c multiplying the x as numbers, how would we operate around ax - cx? It would be the same as (a - c) times the *x*,

(just as it would be if it were 5x - 2x = (5 - 2) x = 3x)

So, write it as:

(a - c) x + b = d

Now, subtract the b from both sides

(a -c) x + b - b = d - b

(a -c) x = d - b

Reminding ourselves that *a, b, c*, and *d* are constants, divide both sides by (a - c),

(a - c)x

a - c

=

d - b

a - c

x =

d - b

a - c

There we have our solution to x, with the constants *a, b, c*, and *d*.

The solutions in the above animation are repeated with a more detailed explanation below.

We have the equation 2a - b = c + 4d. Rearrange this equation four different ways to have an equation that shows the value of *a, b, c, d* each on their own.

Let us start with getting an equation for a=...

2a - b = c + 4d

Add b to both sides

2a - b + b= c + 4d + b

2a = c + 4d + b

Divide both sides by 2

a =

c + 4d + b

2

Now, for b=...

2a - b = c + 4d

Subtract 2a from both sides

2a - b - 2a = c + 4d - 2a

-b = c + 4d - 2a

Divide both sides by -1

-b

-1

=

c + 4d - 2a

-1

b =

c + 4d -2a

-1

For c =...

2a - b = c + 4d

Subtract 4d from both sides

2a - b - 4d = c + 4d - 4d

2a - b - 4d = c

c = 2a - b - 4d

For d=...

2a - b = c + 4d

Subtract c from both sides

2a - b -c = c + 4d - c

2a - b -c = 4d

Divide both sides by 4

2a - b - c

4

=

4d

d

2a - b - c

4

= d

d =

2a - b - c

4