Solve equations requiring the distributive property, combining like terms, and multiple inverse operations. Include equations with fractions by multiplying through by the LCD.
A lesson of Algebra 1, in Solving Linear Equations, about 35 minutes.
By the end of this lesson, you will be able to solve equations that need distributing, combining like terms, and several inverse operations, and clear fractions or decimals out of an equation before you solve it.
Tidy Each Side First, Then Unbuild
A two-step equation is already stripped back:
$$3x + 5 = 20$$
Most equations do not arrive that way. They arrive like this:
$$4(x + 3) - 2 = 18$$
There is nothing new to learn here. There is one thing to do first: make each
side as simple as it can be, and only then start undoing operations.
The routine, in order:
Step
What it does
1
Clear the brackets by distributing
Frees the terms that were trapped inside
2
Combine like terms on each side
Leaves at most one variable term and one number per side
3
Undo the addition or subtraction
Same move as a two-step equation
4
Undo the multiplication or division
Leaves the variable alone
5
Check in the original equation
Catches everything above
Steps 3 and 4 are the two-step equation you already know. Steps 1 and 2 are what
turn a messy equation into that one.
Getting Rid of Fractions Before You Start
Fractions are not harder, but they give arithmetic slips more places to hide.
You can remove all of them in one move.
$$\frac{x}{2} + \frac{x}{3} = 5$$
Multiply every term on both sides by the lowest common denominator, which
here is $6$:
The fractions are gone and what remains is an ordinary equation. The one rule
that matters: every single term gets multiplied, including the ones with no
fraction in them, or the sides stop being equal.
The same trick clears decimals. Multiply by $10$, or $100$, until the decimal
points have gone.
Example 1: Brackets and a Spare Number
Here is the whole method on one equation, worked all the way through.
Solve $4(x + 3) - 2 = 18$.
Multiply out the bracket, leaving everything else as it is 4x + 12 - 2 = 18
Combine the two plain numbers on the left 4x + 10 = 18
Subtract 10 from both sides and simplify 4x = 8
Divide both sides by 4 and simplify x = 2
Check it. Put 2 into the left hand side of the original equation 18
This is the one that catches people. The factor being distributed is negative,
so every sign inside the bracket flips.
Solve $20 - 3(2x - 4) = 8$.
Multiply out the bracket, distributing -3 and leaving the 20 alone 20 - 6x + 12 = 8
Combine the two plain numbers on the left 32 - 6x = 8
Subtract 32 from both sides and simplify -6x = -24
Divide both sides by -6 and simplify x = 4
Worked Example 2: Negative Bracket, Your Turn
Solve $15 - 2(3x + 1) = 1$.
Example 3: Clearing the Fractions First
Solve $\dfrac{x}{2} + \dfrac{x}{3} = 5$.
The denominators are $2$ and $3$, so the lowest common denominator is $6$.
Multiply every term on both sides by $6$ and the fractions disappear.
Multiply every term by 6 and simplify each one 3x + 2x = 30
Combine the like terms on the left 5x = 30
Divide both sides by 5 and simplify x = 6
Check it. Put 6 into the left hand side of the original equation 5
Worked Example 3: Fractions, Your Turn
Solve $\dfrac{x}{4} + \dfrac{x}{6} = 5$.
Worked Example 4: Clearing Decimals
Solve $0.2x + 1.5 = 3.1$.
Every number has one decimal place, so multiplying through by $10$ clears them
all at once.
Example 4: Two Correct Routes Through the Same Equation
Sometimes there is a shortcut, and knowing when to take it is worth more than
knowing one fixed procedure.
$$3(2x + 4) = 30$$
Route one, the standard routine. Distribute, then unbuild:
$$6x + 12 = 30$$
$$6x = 18$$
$$x = 3$$
Route two, divide first. The whole left side is $3$ times the bracket, and
the right side divides by $3$ exactly:
$$2x + 4 = 10$$
$$2x = 6$$
$$x = 3$$
Same answer, one line shorter, and no distributing at all.
The second route only works when the right hand side divides cleanly by the
factor outside the bracket. Here $30 \div 3 = 10$, so it does. If the equation
had been $3(2x + 4) = 31$ you would be carrying thirds through the rest of the
problem, and distributing first would be the easier road.
Look at the equation before you start. Choosing the route is part of the work.
Common Mistakes
Mistake 1: Distributing to the first term only
Wrong: $4(x + 3) = 4x + 3$
Right: $4(x + 3) = 4x + 12$
Everything inside the bracket gets multiplied.
Mistake 2: Distributing a positive when the sign is negative
Wrong: $20 - 3(2x - 4) = 20 - 6x - 12$
Right: $20 - 3(2x - 4) = 20 - 6x + 12$
The factor is $-3$, so $-3 \times -4 = +12$.
Mistake 3: Multiplying only some terms when clearing fractions
Every term, on both sides. The right hand side is a term too.
Mistake 4: Starting to unbuild before tidying up
Given $4x + 12 - 2 = 18$, subtracting $12$ next works, but you then have to
handle the $-2$ separately and it is easy to lose. Combine the numbers on each
side first, and there is only ever one thing to undo at a time.
Mistake 5: Checking against your own working
Substitute back into the original equation, not into a line halfway down. If
the mistake is in that line, checking against it will confirm the mistake.
Where This Goes Next
Every linear equation, however long, collapses to the same thing:
$$ax = b$$
Distributing, combining like terms and clearing fractions are all ways of
getting to that line. Once you are there, one division finishes it.
That is worth knowing because of what comes next. When the variable appears on
both sides, there is one extra move at the front, which is to collect the
variable terms onto one side. After that you are back here, in an equation you
can already finish.
It also explains the two odd cases you will meet shortly. If the variable terms
cancel completely you never reach $ax = b$, and what is left is either a true
statement, in which case every number works, or a false one, in which case no
number does.
Check Your Understanding
Five of them, no instructions this time.
Summary
The routine, in order:
Distribute to clear brackets
Combine like terms on each side
Undo the addition or subtraction
Undo the multiplication or division
Check in the original equation
Clearing fractions: multiply every term on both sides by the lowest common
denominator. Clearing decimals: multiply every term by $10$, or $100$, until
they are gone.
Signs: a negative in front of a bracket flips the sign of every term inside.
Before you start: look at the equation. If the right hand side divides
cleanly by the factor outside the bracket, dividing first is shorter than
distributing.