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Undoing the operations in reverse order, and checking by substituting back.
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 equations that take two steps, such as $4x + 3 = 23$ and $3(x + 4) = 27$, by undoing the operations in the reverse of the order they were done: the last thing done to $x$ is the first thing undone. You will write equations from stories about fees, rates and averages, and you will check every answer by putting it back into the equation you started with.
You can solve one-step equations such as $x + 9 = 14$ and $6x = 42$ by doing the opposite operation to both sides. You know that subtraction undoes addition and division undoes multiplication. You can work with negative numbers, fractions and decimals, and you know the order of operations: multiply and divide before you add and subtract, unless parentheses say otherwise. In this lesson you put two of those one-step moves together, in the right order, to solve equations that need two steps.
| Term | What it means |
|---|---|
| Equation | A statement that two expressions are equal, such as $4x + 3 = 23$. |
| Variable | A letter, such as $x$, that stands for an unknown number. |
| Coefficient | The number that multiplies the variable: in $-5x + 2$ the coefficient is $-5$. |
| Constant term | A number on its own, with no variable: in $-5x + 2$ it is 2. |
| Inverse operation | The operation that undoes another: subtraction undoes addition, division undoes multiplication. |
| Two-step equation | An equation that needs two inverse operations to get the variable alone, such as $px + q = r$ or $p(x + q) = r$. |
| Solution | The value of the variable that makes the equation true. |
A two-step equation tells you what was done to an unknown number. In $4x + 3 = 23$, the unknown $x$ was first multiplied by 4, and then 3 was added, and the result was 23. To find $x$, you undo those two operations, and you undo them in the reverse order: the last thing done is the first thing undone.
Every move is made to both sides, because an equation is a balance: as long as both sides change in the same way, they stay equal. Then you check by putting the answer back in: $4 \times 5 + 3 = 23$, so $x = 5$ is right.
Another way: story
Getting dressed, you put on socks and then shoes. Getting undressed, you take off the shoes first and the socks second. Solving an equation is getting $x$ undressed: whatever was put on last comes off first.
Another way: picture
Picture a balance scale. On the left are 4 identical bags, each holding $x$ marbles, plus 3 loose marbles. On the right are 23 loose marbles. Take 3 loose marbles off each side and it still balances: 4 bags against 20 marbles. Split each side into 4 equal groups and one bag balances 5 marbles.
You might ask why you cannot divide by 4 first in $4x + 3 = 23$. You can, but you must divide everything on each side, including the 3: $x + \tfrac{3}{4} = \tfrac{23}{4}$. That is correct, but it brings in fractions that were not there before. Undoing the addition first keeps the numbers whole and the work short.
The mistake to avoid is dividing only part of a side. Writing $x + 3 = 5.75$ after "dividing by 4" is wrong, because the 3 was not divided. Undoing the last operation first avoids that trap completely, because by the time you divide, there is nothing else on that side.
In $12 - 3x = 27$, the term with $x$ is $-3x$, so the coefficient is $-3$. First subtract 12 from both sides: $-3x = 15$. Then divide both sides by $-3$, sign included: $x = 15 \div (-3) = -5$. Check: $12 - 3(-5) = 12 + 15 = 27$.
A common slip is to divide by 3 instead of $-3$. That gives $x = 5$, and the check shows the mistake at once: $12 - 3(5) = -3$, not 27. Treat the sign in front of a term as part of that term. When the constant is being subtracted, as in $5x - 8 = 17$, undo it by adding 8 to both sides, because adding undoes subtracting.
In $3(x + 4) = 27$, the parentheses say that 4 was added first and the sum was multiplied by 3 last. Undo in reverse order: divide both sides by 3 to get $x + 4 = 9$, then subtract 4 to get $x = 5$.
There is a second way. Use the distributive property to multiply out the parentheses: $3x + 12 = 27$. Now it is an equation of the first kind. Subtract 12 to get $3x = 15$, then divide by 3 to get $x = 5$. Both ways give the same answer, as they must. Dividing first is usually quicker when the right side divides evenly; distributing first is safer when it does not.
The same two moves work when the numbers are fractions or decimals. In $\tfrac{x}{6} + 2 = 7$, subtract 2 to get $\tfrac{x}{6} = 5$, then undo the division by multiplying both sides by 6: $x = 30$. In $0.5x - 1.2 = 3.8$, add 1.2 to get $0.5x = 5$, then divide by 0.5 to get $x = 10$. Dividing by 0.5 is the same as multiplying by 2, which is a useful check.
When a coefficient is a fraction such as $\tfrac{2}{3}$, divide by it by multiplying by its reciprocal: $\tfrac{2}{3}x = 8$ gives $x = 8 \times \tfrac{3}{2} = 12$.
Most two-step equations in real life come from a fixed amount plus a repeated amount. A plumber charges a 60-dollar visit fee plus 45 dollars an hour; a bill of 195 dollars gives $45h + 60 = 195$. A phone that starts at 100% battery and loses 8% an hour reaches 36% when $100 - 8h = 36$.
To write the equation: say in words what the unknown is and give it a letter; find the amount that happens once; find the amount that repeats, and multiply it by the letter; set the sum equal to the total. Then solve, and answer the question in words, with units. In the plumber's case, $45h = 135$ and $h = 3$: the job took 3 hours.
Nothing says the unknown has to be on the left. The equation $50 = 8 + 6x$ is the same as $8 + 6x = 50$, because an equals sign works in both directions: if one side equals the other, the other equals the first. Solve it exactly the same way. Subtract 8 from both sides to get $42 = 6x$, then divide both sides by 6 to get $7 = x$, which you can write as $x = 7$.
The order of the terms on one side does not matter either. $8 + 6x$ and $6x + 8$ are the same expression, because addition can be done in any order. Be careful with subtraction, though: $8 - 6x$ is not the same as $6x - 8$. In $8 - 6x$ the coefficient of $x$ is $-6$, so the last step is to divide by $-6$.
You can solve many story problems without writing an equation at all, by working backward: from 195 dollars take off the 60-dollar fee, leaving 135, and divide by 45 dollars an hour to get 3 hours. Look closely and you will see that these are exactly the steps of solving $45h + 60 = 195$: subtract, then divide.
So why write the equation? Because it records the story in a form you can check, and because it keeps working when the story gets harder, with negative numbers, parentheses or unknowns on both sides. The arithmetic and the algebra are the same thinking; the equation just writes it down.
Use these steps for every two-step equation.
How to check. The check is the most important step, and it takes seconds. Substitute into the equation you started with, not a later line, because a mistake in step 2 would carry into every line after it. If the two sides are equal, you are done. If not, look for the three usual slips: undoing in the wrong order, dropping a negative sign, or changing only one side.
Temperatures in degrees Celsius and degrees Fahrenheit are linked by the formula $F = 1.8C + 32$. A doctor's chart says a patient's temperature is $98.6^\circ$F. To find it in Celsius, solve $1.8C + 32 = 98.6$. Subtract 32 from both sides: $1.8C = 66.6$. Divide both sides by 1.8: $C = 37$. So normal body temperature is about $37^\circ$C. The same equation converts a recipe: an oven set to $350^\circ$F gives $1.8C = 318$ and $C \approx 177$, which is why many recipes printed in Celsius say $180^\circ$C.
Snowy tree crickets chirp faster when it is warm. In 1897 the physicist Amos Dolbear published a rule that is still quoted: the temperature in degrees Fahrenheit is about $T = \tfrac{N}{4} + 40$, where $N$ is the number of chirps in one minute. On a $76^\circ$F evening, how fast should a cricket chirp? Solve $\tfrac{N}{4} + 40 = 76$. Subtract 40: $\tfrac{N}{4} = 36$. Multiply by 4: $N = 144$ chirps a minute. Check: $144 \div 4 + 40 = 36 + 40 = 76$.
A class has 90 dollars to spend on a party. The room rental costs 22.50 dollars, and pizzas cost 13.50 dollars each. How many pizzas can they buy? Write $13.50p + 22.50 = 90$. Subtract the rental: $13.50p = 67.50$. Divide: $p = 5$. Five pizzas use up the money exactly. If they find a coupon that takes 1.50 dollars off each pizza, the equation becomes $12p + 22.50 = 90$, so $12p = 67.50$ and $p = 5.625$: still only 5 whole pizzas, with 7.50 dollars left over for drinks.
The most common mistake is undoing in the wrong order, which usually means dividing only the $x$ term. In $6x + 5 = 29$, writing $x + 5 = \tfrac{29}{6}$ forgets that the 5 must be divided too. Undo the constant first.
The second is losing a negative sign. In $7 - 2x = 15$ the coefficient is $-2$. Subtract 7 to get $-2x = 8$, then divide by $-2$ to get $x = -4$.
The third is using the wrong inverse: subtracting when the equation subtracts, as in turning $3x - 4 = 11$ into $3x = 7$. To undo subtracting 4, add 4: $3x = 15$.
The last is skipping the check, or checking in a later line instead of the original equation, where an early mistake cannot be seen.
List what was done to $x$.
$x \to \times 5 \to +8$
Multiplying came first and adding came last.
Subtract 8 from both sides.
$5x + 8 - 8 = 43 - 8$
Subtracting undoes the adding that was done last.
Simplify both sides.
$5x = 35$
The constant is gone from the left side.
Divide both sides by 5.
$\dfrac{5x}{5} = \dfrac{35}{5}, \quad x = 7$
Dividing undoes multiplying and leaves $x$ alone.
Check in the original equation.
$5 \times 7 + 8 = 35 + 8 = 43$
The left side equals 43, the right side, so $x = 7$ is correct.
Name the coefficient and the constant.
$\text{coefficient } -4, \quad \text{constant } +9$
The negative sign belongs to the $x$ term.
Subtract 9 from both sides.
$-4x + 9 - 9 = -11 - 9$
The constant was added last, so it is undone first.
Simplify both sides.
$-4x = -20$
Subtracting 9 from $-11$ moves further below zero.
Divide both sides by $-4$.
$\dfrac{-4x}{-4} = \dfrac{-20}{-4}$
Divide by the whole coefficient, negative sign included.
Simplify the division.
$x = 5$
A negative divided by a negative is positive.
Check in the original equation.
$-4 \times 5 + 9 = -20 + 9 = -11$
Both sides are $-11$, so the solution is right.
Say what the equation means.
$\text{perimeter} = 2(\text{width} + \text{length}) = 41 \text{ ft}$
A rectangle 6.5 feet long has two widths and two lengths, and the fence uses 41 feet.
Divide both sides by 2.
$\dfrac{2(w + 6.5)}{2} = \dfrac{41}{2}, \quad w + 6.5 = 20.5$
Doubling was done last, so it is undone first.
Subtract 6.5 from both sides.
$w + 6.5 - 6.5 = 20.5 - 6.5, \quad w = 14$
Subtracting undoes the adding inside the parentheses.
Now solve again, starting by distributing the 2.
$2w + 13 = 41$
The 2 multiplies both the $w$ and the 6.5.
Subtract 13 from both sides.
$2w + 13 - 13 = 41 - 13, \quad 2w = 28$
This is now an equation of the form $px + q = r$.
Divide both sides by 2.
$\dfrac{2w}{2} = \dfrac{28}{2}, \quad w = 14$
Both methods give the same width, as they must.
Check in the original equation.
$2(14 + 6.5) = 2 \times 20.5 = 41$
The fence length comes out right.
Answer the question in words.
$\text{width} = 14 \text{ ft}$
The garden is 14 feet wide and 6.5 feet long.
Add 3 to both sides.
$\dfrac{x}{5} - 3 + 3 = 4 + 3, \quad \dfrac{x}{5} = 7$
Adding undoes the subtraction that was done last.
Multiply both sides by 5.
Check in the original equation.
To solve $4x + 5 = 21$, what should you do first?
Complete the worked solution of $\dfrac{x}{8} - 13 = -11$.
Say what was done to $x$, in order.
$x \to \div 8 \to -13$
The subtraction was done last, so it is undone first.
Add $13$ to both sides.
$\dfrac{x}{8} = -11 + 13 =$ a
Adding undoes subtracting, and both sides change by the same amount.
Multiply both sides by $8$.
$x =$ x
Multiplying undoes dividing and leaves $x$ on its own.
Check by putting the value back in.
$\text{your } x \div 8 - 13 = -11$
If the left side comes out as the right side, the solution is correct.
Solve $3x - 9 = -6$. Write one equation per line, ending with $x = $ your answer.
3x - 9 = -6
Solve $3 - 2x = 7$. Write one equation per line, ending with $x = $ your answer.
3 - 2x = 7
Joining a rock-climbing gym costs $25$ dollars once, and then $22$ dollars a month. Priya has paid $223$ dollars in all. How much of that went to the monthly charges, and for how many months has she been a member?
Monthly charges: c dollars. Months: m
Solve $5(x + 3) = 35$. Write one equation per line, ending with $x = $ your answer.
5(x + 3) = 35
Jordan's five bowling scores this month have a mean of $82$. Four of them are $90$, $85$, $95$ and $80$. What is the total of all five scores, and what was the fifth score?
Total: t. Fifth score: s
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A group of $6$ friends goes to a movie. Each buys one ticket and one popcorn that costs $5.50$ dollars. Together they spend $105.00$ dollars. How much did each friend spend, and what is the price of one ticket, in dollars?
Each friend: e dollars. One ticket: t dollars.
You can solve a two-step equation and check it. Without looking: in $6x - 5 = 31$, which operation do you undo first, and why that one? What is the first step for $2(x + 7) = 30$?
20. Your turn: solve $\dfrac{x}{5} - 3 = 4$, step 2
$5 \times \dfrac{x}{5} = 5 \times 7, \quad x = 35$
Multiplying undoes dividing by 5.
20. Your turn: solve $\dfrac{x}{5} - 3 = 4$, step 3
$35 \div 5 - 3 = 7 - 3 = 4$
Both sides are 4, so $x = 35$ is correct.