Back to the on-screen lesson ·
Doing the same thing to both sides until the letter stands alone.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you solve an equation in one variable by doing the same thing to both sides until the letter is alone. The idea underneath is a balance: whatever you do to one side you must do to the other, or the two sides stop being equal. Every step is reversible, which is why you can always check by putting the answer back.
You know that an equation says two amounts are equal, and that a solution is a value of the letter that makes it true. You have solved one-step equations such as $x + 7 = 12$ and $4x = 20$ by doing the opposite operation: subtracting undoes adding, and dividing undoes multiplying. You can multiply and divide negative numbers, and in the last lesson you learned to expand a bracket such as $3(x - 2)$ into $3x - 6$. In this lesson you put those pieces together to solve equations that take several steps, including ones with brackets and negative numbers.
| Term | What it means |
|---|---|
| Equation | A statement that two expressions are equal, such as $4x + 9 = 37$. |
| Solution | A value of the variable that makes the equation true. For $4x + 9 = 37$ it is $x = 7$. |
| Inverse operation | The operation that undoes another: subtracting undoes adding, and dividing undoes multiplying. |
| Isolate | Get the variable alone on one side of the equation, as in $x = 7$. |
| Linear equation | An equation where the variable is only multiplied by numbers and added to numbers, never squared. |
| Properties of equality | The rules that you may add, subtract, multiply or divide both sides by the same number (not dividing by $0$) and they stay equal. |
Think of an equation as a balance scale with the same weight on each side. $4x + 9 = 37$ says that four copies of some unknown number, plus $9$, weigh exactly as much as $37$. Solving means finding the unknown.
The rule that makes solving possible is simple: whatever you do to one side, do to the other. Take $9$ off the left pan and $9$ off the right pan, and the scale still balances:
$$4x + 9 - 9 = 37 - 9 \quad\Rightarrow\quad 4x = 28.$$
Now split each side into four equal parts:
$$\frac{4x}{4} = \frac{28}{4} \quad\Rightarrow\quad x = 7.$$
Each move is an inverse operation. The equation was built by taking $x$, multiplying by $4$, then adding $9$. To get back to $x$, undo those steps in the reverse order: subtract $9$ first, then divide by $4$. It is like taking off shoes and socks: the socks went on first, so they come off last.
Because every move keeps the two sides equal, each new line has exactly the same solution as the line before it. So the last line, $x = 7$, is the solution of the first line too. And you can always check it: $4 \times 7 + 9 = 37$.
Another way: picture
Picture the scale: four identical bags and $9$ loose marbles on the left, $37$ marbles on the right. Take $9$ marbles off each side, leaving four bags against $28$ marbles. Share the $28$ equally among four bags: $7$ in each.
Another way: working backward
Say it as a puzzle: "I think of a number, multiply by $4$ and add $9$. I get $37$." Work backward: $37 - 9 = 28$, then $28 \div 4 = 7$.
To decide what to do first, ask how the equation was built from $x$. Read the side with $x$ as a list of operations, in the order you would do them if you knew $x$.
In $6x - 11 = 25$: first multiply by $6$, then subtract $11$. So undo the subtraction first (add $11$ to both sides: $6x = 36$), then undo the multiplication (divide both sides by $6$: $x = 6$).
In $\frac{x}{5} + 3 = 10$: first divide by $5$, then add $3$. Undo the adding first ($\frac{x}{5} = 7$), then undo the dividing by multiplying both sides by $5$: $x = 35$.
You could divide first, too, since any move that treats both sides the same is allowed. But dividing $6x - 11 = 25$ by $6$ first gives $x - \frac{11}{6} = \frac{25}{6}$, with fractions in every term. Undoing in reverse order keeps the numbers whole for as long as possible, which means fewer mistakes.
Solutions can be negative, and the numbers you divide by can be negative. The moves do not change; only the arithmetic needs care.
Solve $15 - 4x = 35$. The $x$ term is $-4x$. Subtract $15$ from both sides: $-4x = 20$. Now divide both sides by $-4$, the whole coefficient with its sign: $x = -5$. A common slip is to divide by $4$ and get $x = 5$. Check: $15 - 4 \times (-5) = 15 + 20 = 35$. It works, so $-5$ is right.
Solutions do not have to be whole numbers either. $2x + 1 = 8$ gives $2x = 7$, so $x = \frac{7}{2}$, or $3.5$. Leave the answer as a fraction or a decimal; do not round it to a whole number just because the question looks simple. The check still works: $2 \times 3.5 + 1 = 8$.
When the equation has a bracket, you have two good choices.
Expand first. For $4(x + 3) = 44$, expand to get $4x + 12 = 44$. Then subtract $12$ ($4x = 32$) and divide by $4$ ($x = 8$).
Divide first. Both sides of $4(x + 3) = 44$ can be divided by $4$ at once, because the left side is $4$ times something: $x + 3 = 11$, so $x = 8$. This is quicker when the number on the right divides evenly.
When the bracket has a minus sign in front, or when there are other terms on the same side, expanding is safer. For $20 - 2(x - 1) = 8$, expand carefully: $-2 \times x = -2x$ and $-2 \times (-1) = +2$, giving $20 - 2x + 2 = 8$. Collect the numbers: $22 - 2x = 8$. Subtract $22$: $-2x = -14$. Divide by $-2$: $x = 7$. Check: $20 - 2(7 - 1) = 20 - 12 = 8$.
Many equations come from a situation described in words. Before any solving, turn the words into an equation, one phrase at a time.
For a dog walker who charges a $\$40$ sign-up fee plus $\$18$ an hour, with a bill of $\$148$, the equation is $18h + 40 = 148$. Now solve it as usual: $18h = 108$, so $h = 6$ hours.
Last, answer the question in words and ask whether the answer makes sense. Six hours of walking for a total of $\$148$ is reasonable. A negative number of hours, or $600$ hours, would tell you that the equation or the arithmetic has gone wrong somewhere.
Why each move is allowed. If two amounts are equal, they stay equal when you add the same number to both, subtract the same number, multiply both by the same number, or divide both by the same number that is not $0$. These are the properties of equality. Every line you write is equivalent to the one before, with the same solution.
How to check. Put your answer into the original equation, not a later line, since a mistake in copying would hide there. Work out each side on its own. If they are equal, you are right. If not, look for the step where the two sides were treated differently, or a sign that flipped by accident.
Doctors in the United States usually read temperatures in degrees Fahrenheit, while most of the world uses Celsius. The two are linked by $F = 1.8C + 32$. A thermometer shows $102.2\,^{\circ}$F; what is that in Celsius? Solve $1.8C + 32 = 102.2$. Subtract $32$ from both sides: $1.8C = 70.2$. Divide by $1.8$: $C = 39$. A normal body temperature is about $37\,^{\circ}$C, which is $98.6\,^{\circ}$F, so this patient has a fever. The same equation tells you that water boiling at $212\,^{\circ}$F is at $100\,^{\circ}$C: $1.8C = 180$, so $C = 100$.
An eighth-grade class needs $\$950$ for a bus to a science museum. They already have $\$350$ from a bake sale and plan to wash cars at $\$8$ each. How many cars do they need to wash? If $n$ is the number of cars, then $350 + 8n = 950$. Subtract $350$ from both sides: $8n = 600$. Divide by $8$: $n = 75$. Check: $350 + 8 \times 75 = 350 + 600 = 950$. Over two Saturdays that is about $38$ cars a day, which tells the class they need plenty of volunteers with buckets.
A club has $\$75$ to spend on a pizza party. Large pizzas cost $\$14$ each, and delivery costs $\$5$ for the whole order. The most pizzas they can buy solves $14p + 5 = 75$. Subtract $5$: $14p = 70$. Divide by $14$: $p = 5$ pizzas, costing exactly $\$75$. If the answer had not been a whole number, say $p = 5.4$, the club would round down to $5$, since they cannot pay for part of a sixth pizza. Solving tells you the exact break-even point; the situation tells you which way to round.
Doing a move to one side only. Subtracting $9$ from the left but not the right breaks the balance, and the new line has a different solution.
Undoing with the same operation. To undo $-11$, add $11$; do not subtract it again.
Dropping the sign of the coefficient. In $-4x = 20$, divide by $-4$, not $4$: $x = -5$.
Expanding only part of a bracket. In $3(x + 5) = 27$, the $3$ multiplies the $5$ too: $3x + 15 = 27$.
Checking in a later line. Always check in the original equation.
Solve $4x + 9 = 37$. Say what was done to $x$.
$x \;\to\; \times 4 \;\to\; +9$
Undo these in reverse order: the $+9$ first.
Subtract $9$ from both sides.
$4x + 9 - 9 = 37 - 9$
The same amount off both sides keeps them equal.
Simplify both sides.
$4x = 28$
$+9 - 9$ is $0$ on the left; $37 - 9 = 28$ on the right.
Divide both sides by $4$.
$x = 28 \div 4 = 7$
Dividing undoes multiplying by $4$.
Check in the original equation.
$4 \times 7 + 9 = 28 + 9 = 37$
The left side equals the right side, so $x = 7$.
Solve $-2(x + 3) = 14$. Expand the bracket.
$-2x - 6 = 14$
$-2 \times x = -2x$ and $-2 \times 3 = -6$.
Add $6$ to both sides.
$-2x - 6 + 6 = 14 + 6$
Adding $6$ undoes the $-6$.
Simplify both sides.
$-2x = 20$
$-6 + 6 = 0$ and $14 + 6 = 20$.
Divide both sides by $-2$.
$\dfrac{-2x}{-2} = \dfrac{20}{-2}$
Divide by the whole coefficient, sign included.
Simplify the division.
$x = -10$
A positive divided by a negative is negative.
Check the left side of the original equation.
$-2(-10 + 3) = -2 \times (-7) = 14$
Work inside the bracket first, then multiply.
Compare with the right side.
$14 = 14$
The sides agree, so $x = -10$ is the solution.
Solve $7x - 2(x - 6) = 47$. Find the factor of the bracket.
$\text{factor} = -2$
The minus sign in front of the bracket belongs to the factor.
Multiply the $x$ inside by $-2$.
$-2 \times x = -2x$
The factor reaches the first term.
Multiply the $-6$ inside by $-2$.
$-2 \times (-6) = 12$
Negative times negative is positive.
Rewrite the equation without the bracket.
$7x - 2x + 12 = 47$
The expression is equivalent, so the solution is unchanged.
Collect the $x$ terms.
$5x + 12 = 47$
$7x - 2x = 5x$.
Subtract $12$ from both sides.
$5x + 12 - 12 = 47 - 12$
Undo the number term first.
Simplify both sides.
$5x = 35$
Only the coefficient is left to undo.
Divide both sides by $5$.
$x = 35 \div 5$
Dividing undoes multiplying by $5$.
Write the solution.
$x = 7$
The variable is isolated.
Check in the original equation.
$7 \times 7 - 2(7 - 6) = 49 - 2 = 47$
The left side equals $47$, the right side.
Say what was done to $x$.
$x \;\to\; \div 4 \;\to\; -3$
Undo the $-3$ first, then the dividing.
Add $3$ to both sides.
$\dfrac{x}{4} = 8$
Adding $3$ undoes subtracting $3$.
Multiply both sides by $4$.
Check in the original equation.
To solve $3x - 10 = 8$, what is the best first move?
Complete the worked solution of $4(9x + 15) = 276$.
Divide both sides by $4$.
$9x + 15 =$ p
The whole bracket was multiplied by $4$, so dividing by $4$ undoes that first.
Subtract $15$ from both sides.
$9x =$ q
The $+15$ comes off next.
Divide both sides by $9$.
$x =$ r
Dividing undoes multiplying by $9$, the first thing done to $x$.
Solve $4x - 7 = -43$. Write one equation per line, ending with $x = $ your answer.
4x - 7 = -43
Solve $2(x - 5) = -20$. Write one equation per line, ending with $x = $ your answer.
2(x - 5) = -20
Solve $\dfrac{x}{3} + 3 = 5$. Type the value of $x$.
Answer:
Solve $5x + 1 = 3x - 11$. One equation per line, ending with $x = $ your answer.
5x + 1 = 3x - 11
A rectangular bulletin board has a perimeter of $104$ inches. Its length is $2$ inches more than $4$ times its width. How wide is it, in inches?
The width is answer inches.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Solve $6(x - 4) = 6$. Write one equation per line, ending with $x = $ your answer.
6(x - 4) = 6
You can solve a linear equation, including one with a bracket. Without looking: what do you do to both sides first in $3x + 7 = 22$, and why is it allowed?
17. Your turn: solve $\dfrac{x}{4} - 3 = 5$., step 3
$x = 32$
Multiplying undoes dividing by $4$.
17. Your turn: solve $\dfrac{x}{4} - 3 = 5$., step 4
$32 \div 4 - 3 = 8 - 3 = 5$
Both sides are $5$, so $x = 32$.