Solving Linear Equations
Mateo says: "I'm thinking of a number. I triple it, add 4, and get 19. What's my number?" You could guess and check, but there is a method that always works, even when the numbers get ugly. By the end of this page you'll use it, and you'll know Mateo's number.
What an equation says
An equation is a math sentence with an equals sign. It says that two expressions have the same value. Think of a balance scale: whatever is on the left pan weighs exactly the same as what is on the right.
Here the box on the left pan weighs $x$ and sits next to a $3$ lb weight. The right pan holds $10$ lb. Move the sliders and watch when the scale balances.
The scale only stays balanced if you do the same thing to both sides. Take $3$ lb off the left and the scale tips, unless you also take $3$ lb off the right. That is the only rule you need.
To get $x$ alone, use inverse operations, which undo each other:
| Operation in the equation | Undo it with |
|---|---|
| $x + 5$ (add) | subtract $5$ from both sides |
| $x - 5$ (subtract) | add $5$ to both sides |
| $5x$ (multiply) | divide both sides by $5$ |
| $\dfrac{x}{5}$ (divide) | multiply both sides by $5$ |
You may hear the shortcut "move a term to the other side and change its sign." It is just a quick way to say "add or subtract the same number on both sides." Use whichever you prefer, but know why it works.
Look at the following balance scale which is in equilibrium. On the left pan there is a box with an unknown weight $x$ and a $2$ lb weight. On the right pan there is a…
Look at the following balance scale which is in equilibrium. On the left pan there is a box with an unknown weight $x$ and a $2$ lb weight. On the right pan there is a $7$ lb weight. Write the equation that represents this situation and find the value of $x$ so that the scale remains balanced.
Check if the value $x = 4$ is the solution (root) of the following linear equation. To do this, substitute the $x$ with the number $4$ and verify if the equality holds:…
Check if the value $x = 4$ is the solution (root) of the following linear equation. To do this, substitute the $x$ with the number $4$ and verify if the equality holds:
$3x - 5 = 7$
Find the value of $x$ in the following basic equation using the rules for rearranging terms (a term that is added moves to the other side as subtracted and vice versa):…
Find the value of $x$ in the following basic equation using the rules for rearranging terms (a term that is added moves to the other side as subtracted and vice versa):
$x + 15 = 32$
Two-step equations and translating words
Mateo's puzzle has two operations: he multiplies by $3$ and then adds $4$. To undo them, work backward through the same steps in reverse order: undo the adding first, then the multiplying.
A two-step equation like $ax + b = c$ is solved in two moves: first subtract (or add) $b$ on both sides, then divide both sides by $a$.
To write equations from sentences, learn the key phrases:
| Words | Algebra |
|---|---|
| a number plus $5$; $5$ more than a number | $x + 5$ |
| $3$ less than a number; a number decreased by $3$ | $x - 3$ |
| twice a number; double a number | $2x$ |
| a number divided by $4$ | $\dfrac{x}{4}$ |
| "is", "equals", "gives", "the result is" | $=$ |
So "Three times a number $x$ is equal to $21$" becomes $3x = 21$.
Solve the following linear equation step by step: $2x + 8 = 18$ Hint: First move the number that does not have $x$ to the other side of the equality by changing its sign.
Solve the following linear equation step by step:
$2x + 8 = 18$
Hint: First move the number that does not have $x$ to the other side of the equality by changing its sign.
Translate the following sentence into algebraic language and solve the resulting equation: "Three times a number $x$ is equal to $21$".
Translate the following sentence into algebraic language and solve the resulting equation: "Three times a number $x$ is equal to $21$".
In a video game shop, all controllers have the same price. If a group of friends has bought $4$ controllers and they have paid a total of $\$120…
In a video game shop, all controllers have the same price. If a group of friends has bought $4$ controllers and they have paid a total of $\$120$, what is the price of a single controller? Write the corresponding linear equation and calculate the unknown.
Parentheses and variables on both sides
Many equations have parentheses, or $x$ on both sides. The plan is to simplify each side, then collect.
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Distribute to remove parentheses: $3(x + 2) = 3x + 6$.
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Combine like terms on each side.
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Collect the $x$ terms on one side and the plain numbers on the other, using inverse operations.
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Divide by the coefficient of $x$.
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Check in the original equation.
The two sides of an equation are two functions, and the solution is where their values are equal. Graph $y = 5x - 3$ and $y = 2x + 12$: the lines cross at $x = 5$.
When the numbers outside multiply a whole group, you can sometimes divide first. A store takes $\$5$ off each shirt, and $6$ shirts cost $\$90$. The original price $x$ satisfies $6(x - 5) = 90$. Dividing by $6$ gives $x - 5 = 15$, so $x = 20$, and the original price was $\$20$.
Nested parentheses. Work from the inside out, simplifying the innermost group first.
Find the value of the unknown $x$ in the following equation with parentheses, correctly applying the distributive property and the rules of signs:…
Find the value of the unknown $x$ in the following equation with parentheses, correctly applying the distributive property and the rules of signs:
$3(2x - 4) - 2(x + 5) = 14$
In a sporting goods store, the price of a soccer jersey is $\$x$. During the sale, a discount of $\$12$ is applied. A team buys $8$ discounted jerseys for a total of…
In a sporting goods store, the price of a soccer jersey is $\$x$. During the sale, a discount of $\$12$ is applied. A team buys $8$ discounted jerseys for a total of $\$384$. What was the original price of the jersey before the discount?
Solve the following equation containing nested parentheses: $5[2x - 3(x + 1)] = 4[3x - 2(4x - 5)] - 2$
Solve the following equation containing nested parentheses:
$5[2x - 3(x + 1)] = 4[3x - 2(4x - 5)] - 2$
Equations with fractions
Fractions make equations messy, so get rid of them. Multiply both sides by the least common denominator (LCD), the least common multiple (LCM) of all the denominators. Every denominator divides into it evenly, so every fraction disappears.
Finding a common denominator is the same idea as finding equivalent fractions:
Solve the following equation with denominators by first simplifying the expression by calculating the least common multiple:…
Solve the following equation with denominators by first simplifying the expression by calculating the least common multiple:
$\frac{x - 3}{2} - \frac{x + 1}{4} = \frac{5}{4}$
Solve the following linear equation, making sure to correctly operate the negative signs in front of the fractions: $\frac{2x - 1}{3} - \frac{x + 2}{6} = \frac{x}{2}$
Solve the following linear equation, making sure to correctly operate the negative signs in front of the fractions:
$\frac{2x - 1}{3} - \frac{x + 2}{6} = \frac{x}{2}$
Solve the following linear equation step by step, simplifying the denominators using the least common multiple and carefully expanding the parentheses:…
Solve the following linear equation step by step, simplifying the denominators using the least common multiple and carefully expanding the parentheses:
$\frac{3(x - 2)}{4} - \frac{2(x + 3)}{3} = \frac{x - 1}{12} - \frac{5x - 4}{2}$
From a story to an equation
Word problems are the reason equations exist. The skill is translating. Follow the same five steps every time:
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Read the problem and decide what you are asked to find.
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Define the variable: "Let $x$ = ..." with units.
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Write an equation using the facts in the story.
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Solve it.
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Answer the question in a sentence and check it against the story.
When one quantity is compared to others, let $x$ be the one the others are compared to (often the smallest). Then the others become simple expressions like $x + 12$ or $2x$.
A simpler story: Olivia's playlist had some songs, she added $18$, and now it has $50$. With $x$ = original songs, $x + 18 = 50$, so $x = 32$ songs.
Sophia is organizing her library. If today she bought $12$ new books and now has a total of $45$ books, how many books did she have before the purchase? Set up an…
Sophia is organizing her library. If today she bought $12$ new books and now has a total of $45$ books, how many books did she have before the purchase? Set up an equation calling $x$ the initial number of books and solve it.
In an online streaming service, the price of a standard monthly subscription is $\$x$. The "Premium" subscription costs $\$6$ more than the standard one, and the "Basic"…
In an online streaming service, the price of a standard monthly subscription is $\$x$. The "Premium" subscription costs $\$6$ more than the standard one, and the "Basic" subscription costs half as much as the standard one. If a group of friends pays a total of $\$54$ to purchase one subscription of each type, set up and solve the equation to find the price of the standard subscription.
Three siblings share $\$130$. The eldest receives twice as much as the middle one, and the latter receives $\$10$ more than the youngest. Set up an equation where $x$ is…
Three siblings share $\$130$. The eldest receives twice as much as the middle one, and the latter receives $\$10$ more than the youngest. Set up an equation where $x$ is the amount the youngest sibling receives and find out how much money each one receives.
Geometry problems: perimeter
When the story is a shape, draw it and label every side, then use a perimeter formula. The perimeter is the distance around.
For a composite shape, label each side with its expression and add them all. There is also a shortcut. If a rectangle has a rectangular piece cut out of one corner, the perimeter is the same as the full rectangle around it. The notch moves edges inward, but the total length stays the same.
The perimeter of a square measures $24$ in. If we call $x$ the length of one of its sides, write the equation that allows you to calculate this side and find its value.
The perimeter of a square measures $24$ in. If we call $x$ the length of one of its sides, write the equation that allows you to calculate this side and find its value.
Look at the following rectangle. Knowing that its perimeter is $34\text{ cm}$, set up a linear equation and calculate the lengths of its sides and its area.
Look at the following rectangle. Knowing that its perimeter is $34\text{ cm}$, set up a linear equation and calculate the lengths of its sides and its area.
Calculate the value of $x$ so that the perimeter of the following composite shape (a rectangle with a square cut-out in one corner) is $84\text{ cm}$. Note that all…
Calculate the value of $x$ so that the perimeter of the following composite shape (a rectangle with a square cut-out in one corner) is $84\text{ cm}$. Note that all measurements are expressed in terms of $x$.
Organizing information in a table
When a story has several people or items, or a "before and after," build a table first. Then the equation almost writes itself.
A mother's age is three times that of her son. $8$ years ago, the sum of their ages was $44$ years. What is the current age of each one? Organize the data in a table…
A mother's age is three times that of her son. $8$ years ago, the sum of their ages was $44$ years. What is the current age of each one? Organize the data in a table before setting up the equation.
Two water tanks have different capacities. Tank $A$ contains three times as many gallons as tank $B$. If we extract $20$ gallons from tank $A$ and add $10$ gallons to…
Two water tanks have different capacities. Tank $A$ contains three times as many gallons as tank $B$. If we extract $20$ gallons from tank $A$ and add $10$ gallons to tank $B$, tank $A$ will still have twice as much water as $B$. How many gallons were initially in each tank?
In an electronics store, a user buys three types of peripherals: mice, keyboards and headphones. The price of a keyboard is twice that of a mouse, and the headphones are…
In an electronics store, a user buys three types of peripherals: mice, keyboards and headphones. The price of a keyboard is twice that of a mouse, and the headphones are $15\text{ \$}$ more expensive than the keyboard. If they buy $3$ mice, $2$ keyboards and $1$ pair of headphones, and the total bill with an applied discount of $10\text{ \$}$ is $175\text{ \$}$, what is the price of each item?
Equal expressions and number puzzles
Sometimes a story says that two different ways of getting a total end up equal, like two plans costing the same. Write one expression for each plan and set them equal.
Number puzzles use special phrases:
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"$A$ exceeds $B$ by $15$" means $A = B + 15$ (or $A - B = 15$).
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"Twice the difference between a number and $10$" means $2(x - 10)$. The parentheses matter.
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Consecutive integers: $x,\; x + 1,\; x + 2$. Consecutive even (or odd) integers: $x,\; x + 2,\; x + 4$.
A sustainable courier company uses electric scooters for its deliveries. The monthly maintenance cost of a scooter is $45\text{ \$}$, plus $0,15\text{ \$}$ for each mile…
A sustainable courier company uses electric scooters for its deliveries. The monthly maintenance cost of a scooter is $45\text{ \$}$, plus $0,15\text{ \$}$ for each mile travelled. Last month, the total cost of one scooter was equal to that of another model whose fixed maintenance is $60\text{ \$}$, but only costs $0,10\text{ \$}$ per mile. Calculate how many miles both scooters travelled for the expense to be the same.
Find a number such that the sum of its half, its third and its quarter exceeds by $15$ units twice the difference between this number and $10$. Set up the corresponding…
Find a number such that the sum of its half, its third and its quarter exceeds by $15$ units twice the difference between this number and $10$. Set up the corresponding equation and solve it.
Find three consecutive integers such that the difference between the square of the largest and the square of the smallest is equal to $96$. Hint: Remember that…
Find three consecutive integers such that the difference between the square of the largest and the square of the smallest is equal to $96$.
Hint: Remember that $(x+1)^2 = x^2 + 2x + 1$. Although squares appear, you will see that the resulting equation is linear.
Back to the start: Mateo's number is $5$, because $3(5) + 4 = 19$.