Solving Linear Equations
Jayden keeps his trading cards in $2$ identical packs, and he has $3$ loose cards in his pocket. He has $15$ cards in all. How many cards are in each pack? You just set up an equation without noticing: $2x + 3 = 15$. Solving it means finding $x$ without tipping the "scale" out of balance.
What is an equation?
To check a solution, plug the number in for the variable and see whether both sides give the same value. For $x = 6$: $2 \cdot 6 + 3 = 15$ ✓. For $x = 5$: $2 \cdot 5 + 3 = 13 \neq 15$ ✗, so $5$ is not a solution.
Checking if a number is the solution to an equation is essential. State whether the value $x = 4$ is the solution to the following linear equations: a) $x + 7 = 11$ b)…
Checking if a number is the solution to an equation is essential. State whether the value $x = 4$ is the solution to the following linear equations:
- a) $x + 7 = 11$
- b) $3x = 12$
- c) $2x - 1 = 5$
Writing an equation from words
Most of the work in word problems is translating. Read the sentence in small pieces and turn each piece into math.
| Words | Math |
|---|---|
| plus, more than, increased by, added to | $+$ |
| minus, less than, decreased by, taken away | $-$ |
| times, twice, product of | $\times$ (twice $x$ is $2x$) |
| divided by, split equally among | $\div$ |
| is, equals, gives, results in | $=$ |
A quick drawing like this bar model helps you see which parts add up to the whole.
The language of equations allows us to solve mysteries. Imagine you are playing "Guess my number" with a friend. Your friend tells you: "If you add $12$ to my secret…
The language of equations allows us to solve mysteries. Imagine you are playing "Guess my number" with a friend. Your friend tells you:
"If you add $12$ to my secret number, the result is $35$."
Translate this sentence into algebraic language, letting $x$ be the secret number, and write the corresponding equation.
In your favorite video game, you have achieved a score of $x$ points. After completing a daily quest, you were awarded $25$ extra points, and now you have a total of…
In your favorite video game, you have achieved a score of $x$ points. After completing a daily quest, you were awarded $25$ extra points, and now you have a total of $110$ points.
Write the equation that describes this situation and figure out mentally how many points you had at the beginning.
Three friends went out for a snack and ordered three identical milkshakes. When paying with a $\$20$ bill, they received $\$11$ in change. If we call $x$ the price of a…
Three friends went out for a snack and ordered three identical milkshakes. When paying with a $\$20$ bill, they received $\$11$ in change.
If we call $x$ the price of a single milkshake, set up an equation that represents the total cost of the snack plus the change received.
(Hint: $3 \cdot (\text{precio batido}) + \text{vuelta} = \text{total pagado}$)
The balance: isolating the variable
Picture a scale in balance. If you take the same amount off both pans, or put the same amount on both, it stays balanced. That is the golden rule of equations: whatever you do to one side, do to the other side.
To undo an operation, use its inverse operation:
The idea is simple. The variable got "wrapped" in operations, so you unwrap it in the reverse order. In $2x + 3 = 15$, $x$ was multiplied by $2$ and then $3$ was added. So first subtract $3$, then divide by $2$.
One-step equations work the same way: $x - 7 = 12$ gives $x = 12 + 7 = 19$. $5x = 40$ gives $x = 8$. $\dfrac{x}{4} = 6$ gives $x = 6 \times 4 = 24$.
An equation is like a scale in perfect balance. Look at the following image where the orange circles represent our variable $x$ and the gray squares represent units of…
An equation is like a scale in perfect balance. Look at the following image where the orange circles represent our variable $x$ and the gray squares represent units of $1$ lb.
Write the equation that represents the balance of the scale:
Solve these simple linear equations using inverse operations (if an amount is added, subtract it; if an amount is multiplied, divide it): a) $x + 5 = 15$ b) $x - 9 = 2$…
Solve these simple linear equations using inverse operations (if an amount is added, subtract it; if an amount is multiplied, divide it):
- a) $x + 5 = 15$
- b) $x - 9 = 2$
- c) $4x = 24$
- d) $\frac{x}{3} = 5$
On a precision digital scale, we have placed two identical supply boxes (which we will call $x$) and a $25$-gram weight on the left pan. To balance the scale, we have…
On a precision digital scale, we have placed two identical supply boxes (which we will call $x$) and a $25$-gram weight on the left pan. To balance the scale, we have put a $135$-gram weight on the right pan.
Set up the equation that represents this balance and calculate how much each box $x$ weighs.
Writing and solving a story
Now put the two skills together. Follow four steps every time:
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Define the variable (say exactly what $x$ means, with units).
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Write the equation from the story.
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Solve it with inverse operations.
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Check in the story, not only in the equation. Does the answer make sense?
Emma has a collection of trading cards. Her little brother has taken $8$ cards from her and now she only has $22$ left. If we call $x$ the initial number of cards Emma…
Emma has a collection of trading cards. Her little brother has taken $8$ cards from her and now she only has $22$ left.
If we call $x$ the initial number of cards Emma had, write the equation that represents this problem and solve it to find out how many cards she had at the beginning.
A famous streamer is giving away gift cards to his subscribers. He announced that "three times the cards he gave away yesterday, minus the 12 he kept for today, equals…
A famous streamer is giving away gift cards to his subscribers. He announced that "three times the cards he gave away yesterday, minus the 12 he kept for today, equals 48 cards."
If we call $x$ the number of cards he gave away yesterday, write the equation that describes the situation and solve it to find that number.
Perimeter problems
Geometry gives you ready-made equations. The perimeter is the distance around a figure, so add up all the sides. For a rectangle with length $\ell$ and width $w$: $P = 2\ell + 2w$.
For a triangle, just add the three sides. If the sides are $x$, $x + 3$ and $2x - 1$ and the perimeter is $22$ feet, then $x + (x + 3) + (2x - 1) = 22$. That gives $4x + 2 = 22$, so $x = 5$, and the sides are $5$, $8$ and $9$ feet.
The perimeter of a rectangular plot is $60$ feet. We know that the length of the plot is $20$ feet. We want to find out the width ($x$). Use the formula for the…
The perimeter of a rectangular plot is $60$ feet. We know that the length of the plot is $20$ feet. We want to find out the width ($x$).
Use the formula for the perimeter of a rectangle to write an equation:
$2 \cdot 20 + 2 \cdot x = 60$
Write the steps to solve for $x$ and calculate the width of the plot.
We are designing a level for a video game and we need a rectangular platform to have a total perimeter of $50$ feet. The design requires the platform to be $5$ feet…
We are designing a level for a video game and we need a rectangular platform to have a total perimeter of $50$ feet. The design requires the platform to be $5$ feet longer than it is wide.
Calculate the dimensions (width and length) of the platform by setting up an equation where $x$ is the width.
We are designing a new skatepark with a triangular shape. The total perimeter of the track must be $54$ feet. The design conditions are as follows: The longest side…
We are designing a new skatepark with a triangular shape. The total perimeter of the track must be $54$ feet. The design conditions are as follows:
-
The longest side measures $x$ feet.
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The medium side measures half of the long side plus $3$ feet.
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The shortest side measures $10$ feet less than the long side.
Find the dimensions of the three sides of the track by solving the corresponding equation.
Several amounts, one variable
Often a problem describes several quantities that depend on one another. Pick the smallest or "base" amount as $x$ and write the other amounts using $x$.
Consecutive integers are whole numbers in a row, like $14, 15, 16$. If the first is $x$, the next ones are $x + 1$ and $x + 2$.
Three friends are collecting trading cards from an animated series. Lucy has twice as many cards as Mark, and Paula has 10 cards more than Mark. If among the three of…
Three friends are collecting trading cards from an animated series. Lucy has twice as many cards as Mark, and Paula has 10 cards more than Mark. If among the three of them they have a total of $110$ cards, how many does each one have?
Hint: Let $x$ be Mark's cards.
The sum of three consecutive integers is equal to $72$. Set up a linear equation to find what those three numbers are.
The sum of three consecutive integers is equal to $72$. Set up a linear equation to find what those three numbers are.
Three friends decide to buy an annual subscription to a streaming service that costs a total of $\$165…
Three friends decide to buy an annual subscription to a streaming service that costs a total of $\$165$. Based on their usage, they have agreed that:
-
Sarah will pay twice as much as Lucas.
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Lucas will pay $\$15$ more than Matthew.
If we call $x$ the money that Matthew pays, set up a linear equation and calculate how much each person must contribute to complete the payment.
Equations with parentheses
When an equation has parentheses, start by removing them. Use the distributive property: multiply every term inside by the number in front. If the variable ends up on both sides, subtract the smaller variable term from both sides so it appears only once.
If parentheses are nested, work from the inside out. For example, in $2[x - (3 - x)]$, first simplify $3 - x$ inside, remove the inner parentheses, combine, and only then multiply by $2$.
Solve the following linear equation by first removing parentheses and then combining terms with the variable: $4(x - 3) + 2x = 3(x + 2) + 9$
Solve the following linear equation by first removing parentheses and then combining terms with the variable:
$4(x - 3) + 2x = 3(x + 2) + 9$
In a role-playing game, your character's strength is calculated using a "balance formula." To level up, you must find the value of the unknown $x$ that makes both sides…
In a role-playing game, your character's strength is calculated using a "balance formula." To level up, you must find the value of the unknown $x$ that makes both sides of the magic scale equal. Solve the following equation:
$2(3x - 5) - 4(x - 2) = 16 - (x + 3)$
Find the value of $x$ that solves the following complex equation with nested parentheses: $3[2x - (x + 4)] = 2(3x + 1) - 5$ Make sure to solve the inner parentheses…
Find the value of $x$ that solves the following complex equation with nested parentheses:
$3[2x - (x + 4)] = 2(3x + 1) - 5$
Make sure to solve the inner parentheses first and simplify each side before combining terms.
Equations with fractions
If the variable is divided in several places, clear the denominators by multiplying every term on both sides by the LCM (least common multiple) of the denominators. Then the equation has only whole numbers.
Some problems write the LCM of $3$ and $6$ as m.c.m.(3, 6). It is just the LCM, and it works the same way.
Find a number such that half of the number plus one-fourth of the number is equal to $15$. To solve it, find the least common multiple of the denominators to clear the…
Find a number such that half of the number plus one-fourth of the number is equal to $15$. To solve it, find the least common multiple of the denominators to clear the equation of fractions.
$\frac{x}{2} + \frac{x}{4} = 15$
Solve the following equation with fractions, simplifying each step: $\frac{2x - 1}{3} - \frac{x + 2}{6} = 2$ Remember to multiply all the terms of the equation by…
Solve the following equation with fractions, simplifying each step:
$\frac{2x - 1}{3} - \frac{x + 2}{6} = 2$
Remember to multiply all the terms of the equation by $m.c.m.(3, 6)$ to work only with integers.
Solve the following equation with fractions, paying close attention to the negative sign preceding the second fraction:…
Solve the following equation with fractions, paying close attention to the negative sign preceding the second fraction:
$\frac{x + 5}{2} - \frac{2x - 3}{3} = \frac{5}{6}$
Remember that to clear the denominators you must multiply all the terms by their LCM.
Harder stories
Some stories lead to an equation with the variable on both sides, or to fractions of a total. The same four steps work: define, write, solve, check.
When a shape changes ("increase the width by $2$ and decrease the length by $4$"), draw the new rectangle and write its sides with $x$ before you use the perimeter formula.
The age of a famous influencer is three times what it was $10$ years ago. If we call $x$ his current age, can you find out how old he is by setting up and solving a…
The age of a famous influencer is three times what it was $10$ years ago. If we call $x$ his current age, can you find out how old he is by setting up and solving a linear equation?
Your school's robotics team has built a rectangular track for a competition. The length of the track is three times its width. If we increase the width by $2$ feet and…
Your school's robotics team has built a rectangular track for a competition. The length of the track is three times its width. If we increase the width by $2$ feet and decrease the length by $4$ feet, the perimeter is still $44$ feet. Calculate the original dimensions of the track.
A video game store sold a third of the consoles it had in stock during the morning. In the afternoon, it sold half of the remaining consoles. If at closing time they…
A video game store sold a third of the consoles it had in stock during the morning. In the afternoon, it sold half of the remaining consoles. If at closing time they still have $12$ unsold consoles, how many consoles did they start the day with?
Set up the equation letting $x$ be the total number of consoles at the start of the day.
Back to the start: To find how many cards are in each pack, solve $2x + 3 = 15$. Subtract $3$ from both sides to get $2x = 12$, then divide by $2$ to get $x = 6$. Each pack has $6$ cards.