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Substituting the answer back, and what it means when the two sides disagree.
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 check a claimed solution by putting it back into the original equation and seeing whether the two sides come out equal. It takes seconds and catches almost every slip. It is also the thing that makes algebra self-correcting: you never have to wonder whether you got it right, because you can always find out.
You can evaluate an expression: to find $3x + 4$ when $x = 5$, you replace $x$ with $5$ and work out $3 \times 5 + 4 = 19$. You know the order of operations, so you multiply before you add. You can work with negative numbers, and in the last lesson you solved equations such as $4x + 9 = 37$ by doing the same thing to both sides. This lesson is about the last line of every one of those solutions: putting the answer back into the equation to prove that it works. It is also about what to do when it does not work.
| Term | What it means |
|---|---|
| Substitute | Replace a letter with a number everywhere it appears in an expression or equation. |
| Evaluate | Work out the value of an expression once every letter has a number. |
| Solution | A value that makes both sides of an equation equal. |
| Left side and right side | The two expressions on either side of the equals sign. A check works each one out separately. |
| Claim | An answer someone says is correct, before it has been checked. |
| Reasonable | An answer that fits the situation: the right size, sign and units. |
An equation is a statement that two expressions are equal. A value of $x$ is a solution exactly when putting it in makes that statement true. So you never have to wonder whether an answer is right: you can test it.
To check whether $x = 6$ solves $5x - 8 = 22$:
Now try $x = 5$: $5(5) - 8 = 17$, and $17 \ne 22$. So $5$ is not a solution. The check does more than say yes or no. It shows how far off an answer is: $17$ is $5$ short of $22$.
When $x$ appears on both sides, work out each side on its own, with the same value, and then compare the two numbers. Never move terms around during a check; the whole point is to test the equation exactly as it was given.
Checking is fast. Solving an equation might take five lines, but checking the answer takes one. That is why a check belongs at the end of every solution.
Another way: balance
On a balance scale, a solution is the weight for each bag that makes the two pans level. A check puts that weight in the bags and looks at the pans.
Another way: table
Try values in a table: for $5x - 8$, $x = 4$ gives $12$, $x = 5$ gives $17$, $x = 6$ gives $22$. Only $x = 6$ matches the right side, $22$.
Most mistakes in a check happen while substituting, not while solving. Three habits prevent them.
Use brackets. Write the value in brackets, especially if it is negative. For $x = -4$ in $3x + 10$, write $3(-4) + 10$, not $3 - 4 + 10$. The first gives $-12 + 10 = -2$; the second is a different calculation.
Follow the order of operations. Multiply before you add or subtract. In $2 + 5x$ with $x = 3$, the value is $2 + 15 = 17$, not $7 \times 3 = 21$.
Replace every $x$. If $x$ appears twice, both copies get the value. In $6x - 2(x + 1)$ with $x = 3$: $6(3) - 2(3 + 1) = 18 - 8 = 10$.
Work out the brackets first when there are some. For $4(x - 7)$ with $x = 2$, the bracket is $2 - 7 = -5$, and then $4 \times (-5) = -20$. Writing each small result down, rather than doing it all in your head, keeps the signs straight.
If the two sides come out different, the claimed answer is not a solution. That is useful information, not a failure. There are two things to do.
First, check the check. Substitute again, slowly, with brackets. It is possible the answer was right and the check had the slip.
Then find the mistake in the solution. Go through the solution line by line. Every line of a correct solution has the same solution as the original equation. So solve again carefully to find the true answer, then test it in each line of the wrong work. The first line where it fails is the line with the mistake.
For example, suppose someone solved $2x + 6 = 20$ by writing $2x = 26$, then $x = 13$. The check fails: $2(13) + 6 = 32$, not $20$. The true solution is $x = 7$. In the line $2x = 26$, $x = 7$ gives $14 \ne 26$, so that line is the one with the error: $6$ was added to $20$ instead of subtracted.
An equation can be solved correctly and still give an answer that does not fit the question. So after the substitution check, do a second check: is the answer reasonable?
The two checks catch different mistakes. Substituting catches arithmetic slips while solving. Asking whether the answer makes sense catches mistakes in writing the equation in the first place, which substitution cannot catch, because a wrong equation can still be solved correctly.
Substituting is not only for checking a finished answer. You can also use it to find a solution, by guessing, checking and improving the guess. This is slower than solving, but it works even before you know the algebra, and it shows why a solution is special.
Solve $7x + 5 = 61$ by guessing. Try $x = 10$: $7(10) + 5 = 75$, which is $14$ too big. Try a smaller value, $x = 6$: $7(6) + 5 = 47$, which is $14$ too small. The answer is between $6$ and $10$. Try $x = 8$: $7(8) + 5 = 61$. The sides agree, so $x = 8$.
Notice how the left side changed: each time $x$ went up by $1$, the left side went up by $7$, the coefficient. That is why a linear equation has just one solution. The left side climbs steadily, so it passes $61$ exactly once. Guessing is also a quick way to check a solution that looks strange: if an answer of $x = 800$ seems too big, try a small value such as $x = 10$ and see which way you are off.
Why this works. An equation is a claim that two amounts are equal. A solution is, by definition, a number that makes the claim true. So substituting is not a shortcut or a trick: it is testing the definition directly.
Why the check is trustworthy. It uses only evaluation, which is simpler than solving. It does not repeat the moves of the solution, so it does not repeat their mistakes. If you solved by subtracting when you should have added, the check does not know that, and it catches it.
An electrician charges a $\$75$ trip fee plus $\$60$ for each hour of work. The bill says $\$285$ for $3.5$ hours. Is it right? Substitute into the rule: $75 + 60 \times 3.5 = 75 + 210 = 285$. The sides agree, so the bill is right. A second bill says $\$345$ for $4$ hours. Check: $75 + 60 \times 4 = 315$, and $315 \ne 345$. The difference is $\$30$, exactly half an hour of work, so the customer can ask whether they were charged for $4.5$ hours by mistake. The check did not just say wrong; it pointed at what went wrong.
In September 1999, NASA's Mars Climate Orbiter was lost as it reached Mars. An investigation found that one team's software gave thruster results in pound-force seconds, an English unit, while the navigation team's software expected newton-seconds, a metric unit. One pound-force is about $4.45$ newtons, so every number was read as about $4.45$ times too small. The spacecraft flew far lower over Mars than planned. A simple check, substituting a known thruster firing into both teams' formulas and comparing, would have shown two sides that did not agree. Engineers now build checks like that into their work, because solving the wrong equation perfectly still gives the wrong answer.
A family's car goes about $30$ miles on a gallon of gas. For a $360$-mile trip, they solve $30g = 360$ and get $g = 12$ gallons. Check: $30 \times 12 = 360$. It works. Then check that it makes sense: their tank holds $14$ gallons, so one full tank is enough, with $2$ gallons to spare. If they had divided the wrong way and got $g = 0.083$, the substitution $30 \times 0.083 \approx 2.5$ would have shown at once that the answer was far off.
Checking in a later line instead of the original. A mistake made while copying would never be found.
Dropping the brackets. $2x$ with $x = -3$ is $2(-3) = -6$, not $2 - 3$.
Adding before multiplying. In $4 + 3x$ with $x = 2$, the value is $10$, not $14$.
Substituting into only one side. When $x$ is on both sides, both sides need the value.
Treating "close" as correct. $21$ is not $22$. If the sides differ at all, the value is not a solution (unless the difference comes from rounding a decimal answer).
Is $x = 6$ a solution of $5x - 8 = 22$? Substitute into the left side.
$5(6) - 8$
Replace $x$ with $6$, in brackets.
Multiply before subtracting.
$30 - 8$
Multiplication comes before subtraction.
Subtract to finish the left side.
$22$
This is the value of the left side when $x = 6$.
Compare with the right side.
$22 = 22$
The two sides agree.
State the result.
$x = 6 \text{ is a solution}$
A value that makes both sides equal is, by definition, a solution.
Is $x = -3$ a solution of $4 - 2x = 3x + 19$? Substitute into the left side.
$4 - 2(-3)$
Brackets keep the minus sign with the $3$.
Evaluate the left side.
$4 + 6 = 10$
$-2 \times (-3) = +6$: negative times negative is positive.
Substitute into the right side.
$3(-3) + 19$
The same value goes in on this side too.
Multiply on the right side.
$-9 + 19$
$3 \times (-3) = -9$.
Add to finish the right side.
$10$
Each side is worked out on its own.
Compare the two sides.
$10 = 10$
Both sides give the same number.
State the result.
$x = -3 \text{ is a solution}$
The check needed no solving at all, only evaluating.
Ana solved $3(x + 4) = 27$ as $3x + 4 = 27$, $3x = 23$, $x = \frac{23}{3}$. Substitute her answer into the original.
$3\left(\tfrac{23}{3} + 4\right)$
A check always uses the original equation.
Work out the bracket.
$\tfrac{23}{3} + \tfrac{12}{3} = \tfrac{35}{3}$
Write $4$ as $\frac{12}{3}$ to add the fractions.
Multiply by $3$ and compare.
$3 \times \tfrac{35}{3} = 35 \ne 27$
The sides disagree, so her answer is wrong.
Expand the bracket correctly.
$3x + 12 = 27$
The $3$ multiplies the $4$ as well as the $x$.
Subtract $12$ from both sides.
$3x = 15$
Undo the number term first.
Divide both sides by $3$.
$x = 5$
Dividing undoes multiplying by $3$.
Check the new answer in the original.
$3(5 + 4) = 3 \times 9 = 27$
This time the left side matches the right side.
Test the true solution in Ana's first line.
$3(5) + 4 = 19 \ne 27$
A correct line would still be true for $x = 5$.
Test it in the original line.
$3(5 + 4) = 27 \text{ is true}$
So the mistake happened between the original and her first line.
Name the mistake.
$3(x + 4) \ne 3x + 4$
She multiplied only the $x$ by $3$, not the $4$.
Substitute into the left side.
$6(2.5) - 4$
Replace $x$ with $2.5$, in brackets.
Multiply before subtracting.
$15 - 4$
$6 \times 2.5 = 15$.
Subtract to finish the left side.
Compare with the right side.
Which value of $x$ solves $8x - 5 = -37$? Check by substituting.
Leo says $x = 13$ solves $7x - 6 = 71$. Complete the worked solution that checks his claim.
Multiply $7$ by Leo's value.
$7 \times 13 =$ p
Substituting means replacing $x$ with the claimed value; multiply first.
Subtract $6$ to finish the left side.
left side $=$ q
This does not match the right side, $71$, so the claim is wrong.
Solve properly: add $6$ to both sides, then divide by $7$.
$x = (71 + 6) \div 7 =$ r
Undo the subtraction first, then the multiplication.
To check a solution you substitute it. Work out $6x - 5$ when $x = -3$.
The expression comes to answer.
Check whether $x = -1$ solves $4x - 5 = 5x + 5$. Work out each side with $x = -1$.
Left side: left. Right side: right.
Dev solved $4x - 8 = 28$ by subtracting $8$ from both sides and got $x = 5$. Check his answer: what does the left side come to with it? Then find the correct solution.
With Dev's answer the left side is left. The correct solution is x = answer.
The equation $5x + k = 41$ has the solution $x = 6$. Substitute $x = 6$ and find $k$.
After substituting, the x term is p, so k = k.
An equipment rental shop charges $16$ dollars a day plus a $48$ dollar cleaning fee. A customer rents a carpet cleaner for $9$ days and is billed $212$ dollars. Is the bill right?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Check whether $x = -4$ solves $4x + 5 = 2x - 9$. Work out each side with $x = -4$.
Left side: left. Right side: right.
You can check a solution by substitution. Without looking: is $x = 4$ a solution of $3x - 5 = 7$, and how do you know without solving it again?
17. Your turn: is $x = 2.5$ a solution of $6x - 4 = 11$?, step 3
$11$
This is the left side when $x = 2.5$.
17. Your turn: is $x = 2.5$ a solution of $6x - 4 = 11$?, step 4
$11 = 11, \text{ so yes}$
The sides agree, so $2.5$ is a solution.