Exercises: Linear equations

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.

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:

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.

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:

1. a) $x + 7 = 11$

1. b) $3x = 12$

1. c) $2x - 1 = 5$

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.

Solve these simple linear equations using inverse operations (if an amount is added, subtract it; if an amount is multiplied, divide it):

1. a) $x + 5 = 15$

1. b) $x - 9 = 2$

1. c) $4x = 24$

1. d) $\frac{x}{3} = 5$

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.

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: $3 \cdot (\text{precio batido}) + \text{vuelta} = \text{total pagado}$)

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.

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.

Solve the following linear equation by first removing parentheses and then combining terms with the variable:

$4(x - 3) + 2x = 3(x + 2) + 9$

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.

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.

$\frac{x}{2} + \frac{x}{4} = 15$

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.

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.

The sum of three consecutive integers is equal to $72$. Set up a linear equation to find what those three numbers are.

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:

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

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.

• 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.

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.

• The medium side measures half of the long side plus $3$ feet.

• 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.

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.

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 decrease the length by $4$ feet, the perimeter is still $44$ feet. Calculate the original dimensions of the track.

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.

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.