Linear equations in one variable

The equation has one unknown letter, the variableA letter that stands for a number you don’t know yet, like the x in 5x + 7.. It's never squared, it's never in the bottom of a fraction, and there's no second unknown.

Read the last line first. It might say:

• “What value of $x$ is the solution to the given equation?”
• “If $7x+3=38$, what is the value of $70x+30$?”
• “Which equation has the same solution as the given equation?”
• “Which equation represents this situation?”
• “How many solutions does the given equation have?” The choices are Zero, Exactly one, Exactly two and Infinitely many.

In the second one, $x$ isn't the answer. The question wants the whole expression $70x+30$.

Some questions add a second letter and call it a constantA number that doesn’t change, like the 7 in 5x + 7, or a letter such as p that stands for one fixed value.. A constant is a number that stays fixed. It can be a plain number, like the 7 in $5x+7$, or a letter like $p$ that stands for one number you haven't been told. The last line might read “In the given equation, $p$ is a constant. The equation has no solution. What is the value of $p$?” Then the answer is $p$, and you never need to find $x$.

Here's one with every move written out: $3(5x-2)=4x+16$.

1. Start.

   $$3(5x-2)=4x+16$$

2. Multiply both terms in the parentheses by 3.

   $$15x-6=4x+16$$

3. Subtract $4x$ from both sides.

   $$11x-6=16$$

4. Add 6 to both sides.

   $$11x=22$$

5. Divide both sides by 11.

   $$x=2$$

6. Check: put 2 into the original. Both sides come out to 24.

   $$3(10-2)=24 \text{ and } 4(2)+16=24$$

In the third row, $15x$ and $4x$ are like termsParts of an expression with the same letter raised to the same power, like 5x and 2x, which you can add together.. Both are a number times $x$, so they combine into $11x$. Every row does the same thing to both sides, which keeps the two sides equal. Undo the adding and subtracting first, then the multiplying.

Most slips happen in the fourth and fifth rows.

Desmos can check any of these. Type 3(5x-2)=4x+16 on its own line, and Desmos draws a vertical line at the solution. Click where it crosses the horizontal axisOne of the two lines a graph is measured against, one running across and one running up. to read $x=2$. Typing takes about ten seconds. If the algebra comes easily to you, use Desmos as a check. If signs and fractions are where you lose points, the ten seconds are worth it every time.

To clear fractions, multiply every term on both sides by a number all the bottoms go into.

1. Start. 5 and 4 both go into 20.

   $$\frac15(x+2)-\frac14(x+2)=-3$$

2. Multiply every term on both sides by 20.

   $$4(x+2)-5(x+2)=-60$$

3. 4 copies of $x+2$ minus 5 copies leaves minus one copy.

   $$-(x+2)=-60$$

4. Multiply both sides by $-1$.

   $$x+2=60$$

5. Subtract 2 from both sides.

   $$x=58$$

Keep $x+2$ in one piece the whole way. Multiplying it out twice takes longer and gives you more places to slip.

Decimals work the same way with 10 or 100. With money, multiply by 100 to work in cents.

1. Start.

   $$0.15x+2.40=9.15$$

2. Multiply every term by 100.

   $$15x+240=915$$

3. Subtract 240 from both sides.

   $$15x=675$$

4. Divide both sides by 15.

   $$x=45$$

Some questions don't want the letter at all. Each kind below has a quicker route than solving.

Which equation has the same solution as the given equation?

• A. $13w=36$
• B. $13w=468$
• C. $13w=486$
• D. $w=468$

Subtract 9 from both sides and you're done. B is correct. C adds 9 instead of subtracting it, and D drops the 13 from one side only.

When the question asks for an expression, compare it with the side you're given. If $8z+6=46$, what is the value of $16z+12$?

1. Start. $16z$ is $2\times8z$, and 12 is $2\times6$.

   $$8z+6=46$$

2. Multiply both sides by 2.

   $$2(8z+6)=2(46)$$

3. Multiply each term by 2.

   $$16z+12=92$$

The multiplier can be bigger. If $7x+3=38$, then $70x+30$ is 10 times the left side, so it's $10(38)=380$. You never have to find $x$.

A group that shows up twice can be solved as one piece. If $9(x-2)=5(x-2)+28$, what is the value of $x-2$?

1. Start. Treat $x-2$ as one thing.

   $$9(x-2)=5(x-2)+28$$

2. Subtract $5(x-2)$ from both sides. 9 copies minus 5 copies is 4 copies.

   $$4(x-2)=28$$

3. Divide both sides by 4.

   $$x-2=7$$

Here's one to try.

Name the unknown in words first, with its unit: “$m$ is the number of months.” Then find the amount that comes once and the amount per unit.

A gym charges a $45 enrollment fee plus $22 a month. Kai has paid $243 in all.

Subtract 45 from both sides to get $22m=198$, then divide by 22. Kai has been a member for 9 months.

A cost comes off once too. A newsletter earns $5 per subscriber and pays $2,400 in fixed costs. For a profit of $8,000, the equation is $5s-2400=8000$, so $s=2{,}080$ subscribers. If you leave out the cost, you get $5s=8000$ and 1,600 subscribers. That equation counts every dollar coming in as profit.

When one amount is a multiple of another, name the smaller one first. A batch of fruit punch is 2,450 milliliters of syrup, juice and soda water. It has 6 times as much juice as syrup, and 350 milliliters of soda water.

1. Add the three amounts.

   $$m+6m+350=2450$$

2. $m+6m=7m$. The syrup counts too.

   $$7m+350=2450$$

3. Subtract 350 from both sides.

   $$7m=2100$$

4. Divide both sides by 7.

   $$m=300$$

The punch has 300 milliliters of syrup.

A crew charges $950 for the first 3 hours of a job and $50 for each hour after that. A job of $h$ hours costs $1,200.

Test it at $h=3$. The extra charge should be zero, and $50(3-3)=0$.

A theater has 14 seats in its front row and 18 in each row behind it, 302 seats in all. That's $14+18r=302$, where $r$ is the number of rows behind the front row.

You don't need to solve to answer this. Decide whether the number multiplies the letter or stands alone, and you know what it means.

Simplify both sides all the way. Then compare the $x$ terms, and after that the numbers on their own.

If you subtract $2x$ and the $x$ terms cancel, read what's left. $10=3$ is never true, so no $x$ works. $10=10$ is always true, so every $x$ works.

A linear equation can't have exactly two solutions.

Here the question gives you the number of solutions and asks for the constant.

In the given equation, $c$ is a constant. The equation has no solution. What is the value of $c$?

1. Multiply out the left side.

   $$12x-20=(7c+2)x+9$$

2. No solution needs matching $x$ terms, so set the numbers multiplying $x$ equal.

   $$12=7c+2$$

3. Subtract 2 from both sides.

   $$10=7c$$

4. Divide both sides by 7.

   $$c=\frac{10}{7}$$

5. Check: the $x$ terms match and $-20$ isn't 9, so no $x$ works.

   $$12x-20=12x+9$$

For infinitely many solutions, the numbers on their own have to match too. Here $-20$ and 9 never match, so no value of $c$ gives infinitely many.

When the $x$ terms are all on one side, collect them first. In this one, $p$ is a constant and the equation has no solution.

1. Pull out the $x$: $6x-18px$ is $(6-18p)x$.

   $$(6-18p)x=45$$

2. The right side has no $x$, so its $x$ part is 0. For no solution, the left $x$ part must be 0 too.

   $$6-18p=0$$

3. Add $18p$ to both sides, then divide by 18. $\frac{6}{18}=\frac13$.

   $$p=\frac13$$

4. Check: the equation becomes $0=45$, which is never true.

   $$0=45$$

Answers like $\frac{10}{7}$ and $\frac13$ are normal here, so don't change your work because the answer isn't a whole number. On a grid-inA question where you type your own answer instead of picking one of four.Read the lesson, type the fraction, such as 10/7. You get five characters for a positive answer and six for a negative one, counting the minus sign. A decimal cut off too short can be marked wrong.

The last version asks what the constant can't be.

In the given equation, $r$ is a positive integer. If the equation has exactly one solution, which of the following CANNOT be the value of $r$?

• A. 2
• B. 5
• C. 7
• D. 10

Exactly one solution needs different $x$ terms. So the one value that fails is the one that makes them match.

1. Multiply both terms in the parentheses by 2.

   $$2rx+6=10x+7$$

2. The $x$ terms match when the numbers multiplying $x$ are equal.

   $$2r=10$$

3. Divide both sides by 2. B is correct.

   $$r=5$$

With $r=5$, the equation is $10x+6=10x+7$, which has no solution. Choice D comes from matching $r$ with 10 before multiplying out the 2.

Do these by hand. A Desmos slider for the constant can show you roughly where the answer is, but it won't give you an exact value like $\frac{10}{7}$.

Before each one, check whether the last line wants $x$, an expression, an equation, a count of solutions or a constant. When you finish, make sure your answer is that thing.