Solving One-Step Equations: Free Response

Read the first equation before touching it. The letter has $12$ added to it, and addition is undone by subtraction, so subtract $12$ from both sides:

$n + 12 - 12 = 5 - 12$

On the left the $+12$ and the $-12$ cancel, leaving the letter alone. On the right the subtraction runs past zero, since $12$ is larger than $5$, so the result is negative:

$n = -7$

Check it in the equation as first written, not in any line you produced along the way: $-7 + 12 = 5$, which is the right side, so the value stands.

The second equation has $4.6$ subtracted from the letter, so the inverse is to add $4.6$ to both sides:

$w - 4.6 + 4.6 = 2.9 + 4.6$

The left collapses to $w$, and the right is a decimal addition with the points lined up:

$w = 7.5$

Check: $7.5 - 4.6 = 2.9$, which is the right side, so this one stands too.

In the first equation the letter is multiplied, and the coefficient is $-9$ with its sign, not a bare $9$. Multiplication is undone by division, so divide both sides by the whole coefficient:

$\frac{-9k}{-9} = \frac{72}{-9}$

On the left, $-9$ divided by $-9$ is $1$, so the letter stands alone. On the right, a positive divided by a negative is negative, and $72 \div 9 = 8$:

$k = -8$

Check: $-9(-8) = 72$, since a negative times a negative is positive, and $72$ is the right side.

In the second equation the letter has already been divided, by $8$, and division is undone by multiplication, so multiply both sides by $8$:

$8 \cdot \frac{h}{8} = 8 \cdot (-2.5)$

The left becomes $h$. On the right, $8 \times 2.5 = 20$, and a positive times a negative is negative:

$h = -20$

Check: $-20 \div 8 = -2.5$, which is the right side. Dividing by $8$ a second time, the kind of move the first equation called for, would have taken the letter further from standing alone rather than closer.

Start with what the line records. Nothing crossed the equals sign, because nothing in an equation can travel: one operation, subtracting $12$, was carried out twice, once on the left side and once on the right. The left side then simplified to the letter alone, which is why the move was worth making, and the right side simplified to a single number. The classmate's description names the appearance of the finished line rather than the act that produced it, and a student who believes things travel has no way to decide what to do when the letter is multiplied instead of added to.

Now why the solutions are the same. Suppose some value of the letter makes the original equation true. With that value in place, the left side and the right side are not merely close, they are one and the same number; call it $N$. Subtracting $12$ from both sides leaves $N - 12$ on the left and $N - 12$ on the right, and one number treated the same way twice cannot come out two different ways:

$N = N \;\Rightarrow\; N - 12 = N - 12$

So the value still makes the new equation true, and no solution has been thrown away.

Nothing has been gained either, and that direction needs the second half of the argument. The move can be undone: adding $12$ to both sides of the new equation returns the original one exactly. So any value making the new equation true makes the original true, by running the argument above on the undoing move. The two equations therefore hold for the same values of the letter, which is what lets you rewrite an equation step by step and trust the answer at the end.

Change the left side only and the first step of that argument fails. The left becomes $N - 12$ while the right is still $N$, and those are two numbers $12$ apart rather than one number written twice. The value that made the original true does not make the new sentence true at all, so the new sentence is no longer a reliable stand-in for the old one.

Substitute each reported value where its letter stood, and evaluate the two sides separately, so that the comparison at the end is between two numbers.

The first equation is $r - 7.5 = 11$ with $r = 3.5$ reported. The left side is a subtraction that runs past zero:

$3.5 - 7.5 = -4$

The right side is $11$, and $-4$ is not $11$, so the two sides do not come out level.

The second equation is $\dfrac{p}{6} = 1.5$ with $p = 9$ reported. The left side is $9 \div 6$, which is $1.5$ because $6 \times 1.5 = 9$:

$\frac{9}{6} = 1.5$

The right side is $1.5$, so this pair matches exactly.

The third equation is $-6c = 39$ with $c = 6.5$ reported. The left side is a negative times a positive, so it is negative:

$-6(6.5) = -39$

The right side is $39$, and $-39$ is not $39$. The two sides are the same size and opposite in sign, which is a strong hint about where the working went astray, but the check itself reports only that they disagree.

Take the subtraction equation first. The letter has $7.5$ subtracted from it, so the inverse is to add $7.5$ to both sides:

$r - 7.5 + 7.5 = 11 + 7.5$

The left collapses to $r$ and the right is a decimal addition:

$r = 18.5$

Check: $18.5 - 7.5 = 11$, which is the right side, so this value survives its own check.

The reported value is what you get by subtracting where you should add, since $11 - 7.5 = 3.5$. That is the commonest slip of all: using the operation the equation shows rather than the one that undoes it.

Now the multiplication equation. The coefficient is $-6$, sign included, so divide both sides by $-6$:

$\frac{-6c}{-6} = \frac{39}{-6}$

A positive divided by a negative is negative, and $39 \div 6 = 6.5$:

$c = -6.5$

Check: $-6(-6.5) = 39$, since a negative times a negative is positive, and $39$ is the right side.

The reported value is what you get by dividing by a bare $6$ and leaving the minus sign behind. It has the right size and the wrong sign, which is exactly what part A's check showed by returning a left side of the same size and the opposite sign.

Everything here follows from the definition. A solution of an equation is a value of the letter that makes the equation a true statement. Put the value in place, work the two sides out separately, and call what they come to $L$ and $R$. The value is a solution exactly when

$L = R$

and a substitution check is nothing more than computing $L$ and $R$ and making that comparison.

So a check that comes out level settles the whole of what the definition asks. The value is a solution, and no further working can overturn that, because there is nothing more to being a solution than the equality the check has just witnessed.

What it leaves open is the route. The definition says nothing about how the number was found, so a value reached by a muddled method, by trying numbers, or by a slip that happened to cancel itself out passes the check exactly as a value reached by the inverse operation does. This is why the classmate's wording is too strong: the check judges the value, not the working. It is also why the check is worth running even when you are confident, since it is the working, not the value, that you have already looked at.

A check that does not come out level settles the matter in the other direction, and settles it with no exceptions. If the two sides evaluate to different numbers, the equation is a false statement at that value, so the value is not a solution. There is no rescuing it later with better presentation.

One addition makes the level check stronger still here. Each of these equations attaches one operation to the letter, so the inverse move produces exactly one value and there is only one solution to find. With only one solution available, a value that passes the check is not merely a solution; it is the one the question wanted.

The evening temperature is the unknown, and the night does one thing to it: it takes $6.8$ degrees away. What is left is the opening reading, so the sentence the situation asserts is

$T - 6.8 = -3.2$

This is the form in which a number is subtracted from the letter, so the inverse is to add that number to both sides:

$T - 6.8 + 6.8 = -3.2 + 6.8$

The left collapses to $T$. On the right, adding $6.8$ to $-3.2$ moves $6.8$ up the number line from a point $3.2$ below zero, which carries it $3.6$ above zero:

$T = 3.6$

So the room was at $3.6$ degrees Celsius at closing time. Check it against the situation as described, which is what the check is for: starting at $3.6$ and falling $6.8$ gives $3.6 - 6.8 = -3.2$, the opening reading.

Cutting the wheel into $16$ equal wedges divides its weight by $16$, and one of those wedges weighs $0.85$ kg, so the wheel's weight satisfies

$\frac{W}{16} = 0.85$

The letter has been divided, so multiply both sides by $16$. Multiplying as whole numbers, $16 \times 85 = 1360$, and two decimal places in the factor put the point two from the right:

$W = 16 \cdot 0.85 = 13.6$

The whole wheel weighed $13.6$ kg. Check: $13.6 \div 16 = 0.85$, one wedge.

Each wedge sold brings in $6.40$ dollars, so $s$ wedges bring in $6.4s$ dollars, and that came to $96$ dollars:

$6.4s = 96$

Here the letter is multiplied, so divide both sides by $6.4$. Moving both points one place makes the divisor whole: $96 \div 6.4 = 960 \div 64 = 15$.

$s = 15$

Check: $6.4 \times 15 = 96$ dollars. Read the two answers back against the situation before accepting them: a count of wedges has to be a whole number, and the wheel has to have been cut into enough wedges to supply it. This one clears both tests.

The situation is unchanged apart from the opening reading, so the equation has the same shape:

$T - 6.8 = -9.4$

The letter still has $6.8$ subtracted from it, so add $6.8$ to both sides. On the right, adding $6.8$ to $-9.4$ climbs $6.8$ from a point $9.4$ below zero, which stops $2.6$ short of zero:

$T = -9.4 + 6.8 = -2.6$

Check: $-2.6 - 6.8 = -9.4$, the opening reading.

The sign is the interpretation. A negative closing temperature means the room was already below zero when the shop shut, so the night did not carry it below zero, it carried it further below. In part A the answer was positive, and the same night was what took the room from above zero to below it.

Why the move never changed: the choice of move is made by reading the left side, where the letter lives, and both versions of the problem show the same attachment, a subtraction of $6.8$. The number on the other side plays no part in that decision. It only decides what arithmetic you do once the move is chosen, and here it decided whether the final addition landed above zero or below it. That is the reason one method covers negative, decimal and fractional cases alike: the method reads the attachment, and the arithmetic handles the numbers.

The letter is divided by $6$, so multiply both sides by $6$. On the right, $6 \times 4.5 = 27$, and a positive times a negative is negative:

$6 \cdot \frac{z}{6} = 6 \cdot (-4.5) \;\Rightarrow\; z = -27$

Check in the original: $-27 \div 6 = -4.5$, which is the right side.

Now the second move. The solved equation is $z = -27$, and multiplying both sides of it by $2$ gives

$2z = -54$

Test the same value in this new equation: $2(-27) = -54$, which is the right side, so the value has survived the move. It is worth noticing that nothing else has been let in either. The new equation is the multiplication form, and dividing both sides by $2$ takes you straight back to $z = -27$, so it accepts that value and no other.

The second move made the equation less useful, not more, which is exactly why the moves you choose while solving run the other way. But usefulness and truth are different questions, and this part is about truth.

Fix an equation and suppose some value of the letter is a solution of it. Substituting that value makes the equation a true statement, which is to say the left side and the right side are not two numbers that happen to be close, they are one number written two ways. Call that number $N$: the left side is $N$ and the right side is $N$.

Add the same number, say $a$, to both sides. The left becomes $N + a$ and the right becomes $N + a$:

$N = N \;\Rightarrow\; N + a = N + a$

One number added to one number gives one answer, so these are the same, and the value is still a solution of the new equation. Multiplying both sides by the same nonzero number $b$ runs identically: the left becomes $N \cdot b$, the right becomes $N \cdot b$, and a single product cannot come out two ways. Subtraction and division are the same argument again, which is why one paragraph covers all four moves.

So far nothing is lost. The remaining worry is the opposite one: the new equation might be true at values the original rejected, in which case solving the new one would hand you an answer that fails the question. Here the undoing move settles it. Adding $a$ to both sides is undone by subtracting $a$ from both sides, and multiplying by a nonzero $b$ is undone by multiplying by its reciprocal, since $b$ times its reciprocal is $1$. Take any value that solves the new equation and apply the undoing move: by the paragraph above, that value solves the equation the undoing move produces, which is the original.

The two directions together say that the original equation and the new one are true at exactly the same values. That is the guarantee the whole method rests on, and it is why a chain of these moves ending in the letter standing alone has the answer on the other side.

Carry the instruction out literally. Multiplying any number by zero gives zero, so the left side becomes $0$ and the right side becomes $0$:

$0 = 0$

Now ask the question the definition of a solution asks: which values of $m$ make that a true statement? Every one of them. The letter has vanished from the sentence, so nothing about $m$ can make the sentence false, and $m = 2.4$, $m = 0$ and $m = -100$ all satisfy it equally well.

Compare that with the equation you started from. Solving $m + 2.6 = 5$ by subtracting $2.6$ from both sides gives $m = 2.4$, and that is the only value the original accepts. So the move has not lost anything, in the sense of part B, but it has let in everything else, and the new statement is no longer a stand-in for the old one. A sentence true of every number tells you nothing about which number you were looking for.

This is the failure the other half of part B was built to prevent, and it fails for the reason that argument named. Every safe move came with an undoing move: adding is undone by subtracting, and multiplying by a nonzero number is undone by multiplying by its reciprocal. Multiplication by zero has no such partner. Undoing it would mean dividing both sides by zero, and division by zero is undefined, so there is no way back from $0 = 0$ to the equation that produced it. Once you cannot walk a move back, the guarantee that the two equations hold at the same values is gone.

So the restriction written on the division move and the trouble with multiplying by zero are one restriction seen twice. Zero is the number with no reciprocal, and the moves that stay safe are precisely the reversible ones.

In the first equation the letter is multiplied by $8$, so divide both sides by $8$. Dividing $4.8$ by $8$ gives $0.6$, since $48 \div 8 = 6$ and $4.8$ is a tenth of $48$:

$\frac{8k}{8} = \frac{4.8}{8} \;\Rightarrow\; k = 0.6$

Check: $8(0.6) = 4.8$, which is the right side.

In the second equation the letter is divided by $8$, so multiply both sides by $8$. As whole numbers $8 \times 48 = 384$, and one decimal place puts the point one from the right:

$8 \cdot \frac{k}{8} = 8 \cdot 4.8 \;\Rightarrow\; k = 38.4$

Check: $38.4 \div 8 = 4.8$, which is the right side.

The two equations look almost the same on the page and their answers are nowhere near each other, which is the point of the next part.

The move is chosen by reading the side the letter is on. In $8k = 4.8$ the letter has been multiplied by $8$, and multiplication is undone by division, so both sides are divided by $8$. In $\dfrac{k}{8} = 4.8$ the letter has already been divided by $8$, and division is undone by multiplication, so both sides are multiplied by $8$. The two equations differ in one operation, and the moves that undo them differ in exactly the same way.

Now the comparison. Both answers are built from the same number, $4.8$, by opposite acts. Dividing gives

$4.8 \div 8 = 0.6$

and multiplying gives

$4.8 \times 8 = 38.4$

Getting from the smaller to the larger therefore means undoing one division by $8$ and then applying one multiplication by $8$, which is multiplying by $8$ twice over:

$8 \times 8 = 64$

So the larger answer is $64$ times the smaller. Dividing $38.4$ by $0.6$ confirms it, but the factor was readable from the two moves before either answer was worked out, and that is the more useful way to see it: a mistake that swaps one of these forms for the other does not miss by a little, it misses by the square of the number attached to the letter.

Test the reported value first, since that costs one line and settles whether there is anything to diagnose. Substituting it into the second equation gives a left side of

$0.6 \div 8 = 0.075$

and the right side is $4.8$. Those are nowhere near each other, so the value is not a solution of the second equation, whatever its merits elsewhere.

Now apply the student's move honestly, because what they wrote down is not what their move produces. Dividing both sides of $\dfrac{k}{8} = 4.8$ by $8$ gives

$\frac{k}{64} = 0.6$

On the left, the letter was already divided by $8$, and dividing by $8$ again is dividing by $8$ twice over, which is dividing by $64$. On the right, $4.8 \div 8 = 0.6$. So the move is legitimate in the sense of part B of the previous question, since the same operation was applied to both sides and the equation it produces is true at the same value, but it is useless: the letter is buried deeper than before, and repeating the move would bury it further.

That is the difference between a move that is allowed and a move that helps. The balance principle says what you may do; the goal of isolating the letter says what is worth doing. Reading the attachment first is what tells the two apart, and the check is what catches it when they are confused, since a value copied from a different equation was never going to survive substitution here.