First-Degree 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.

In your favorite video game, you have achieved a score of $x$ points. After completing a daily quest, you were rewarded with $25$ extra points, and now you have a total of $110$ points.

Write the equation that describes this situation and work out in your head how many points you had at the beginning.

Checking whether 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) $2x - 1 = 5$

An equation is like a balance scale in perfect equilibrium. Look at the following image where the orange circles represent our unknown $x$ and the gray squares represent $1$ kg units.

Write the equation representing the balance of the scale:

Solve these very simple linear equations using inverse operations (if something adds, subtract it; if something multiplies, divide by it):

1. a) $x + 5 = 15$

1. b) $x - 9 = 2$

1. c) $4x = 24$

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

On a precision digital balance, 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 representing this balance and calculate how much each box $x$ weighs.

A famous streamer is giving away gift cards to their subscribers. They announced that "three times the number of cards given away yesterday, minus the 12 saved for today, equals 48 cards".

If we let $x$ be the number of cards given away yesterday, write the equation that describes the situation and solve it to find that number.

Marta has a collection of soccer trading cards. Her little brother took $8$ cards from her, and now she only has $22$ left.

If we let $x$ be the initial number of cards Marta had, write the equation representing this problem and solve it to find out how many cards she had at the start.

The perimeter of a rectangular plot measures $60$ meters. We know that the length of the plot is $20$ meters. We want to find out the width ($x$).

Use the formula for the perimeter of a rectangle to write an equation:

$2 \times 20 + 2 \times x = 60$

Write down the steps to solve for $x$ and calculate the width of the plot.

Solve the following linear equation by first expanding the brackets and then grouping the terms with the unknown:

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

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 of the magic balance 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 brackets:

$3[2x - (x + 4)] = 2(3x + 1) - 5$

Make sure to solve the inner brackets first and simplify each side before rearranging terms.

Find a number such that half of it plus a quarter of it equals $15$. To solve it, find the lowest common multiple of the denominators to clear the equation of fractions.

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

Solve the following equation with denominators, simplifying each step:

$\frac{2x - 1}{3} - \frac{x + 2}{6} = 2$

Remember to multiply all the terms in the equation by $m.c.m.(3, 6)$ to work with integers only.

Solve the following equation with denominators, paying close attention to the negative sign in front of the second fraction:

$\frac{x + 5}{2} - \frac{2x - 3}{3} = \frac{5}{6}$

Remember that to clear the denominators, you must multiply all terms by their LCM.

Three friends went to have a snack and ordered three identical milkshakes. When paying with a $400$ pesos note, they received $220$ pesos in change.

If we let $x$ be the price of a single milkshake, set up an equation that represents the total cost of the snack plus the change received.

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

Three friends are collecting trading cards from an animated series. Lucía has twice as many cards as Marcos, and Paula has 10 more cards than Marcos. If the three of them have a total of $110$ cards altogether, how many does each person have?

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

Three friends decide to buy an annual subscription to a streaming platform that costs a total of $3\,300$ pesos. Based on how much they use it, they have agreed that:

• Sara will pay twice as much as Lucas.

• Lucas will pay $300$ pesos more than Mateo.

If we let $x$ be the amount of money Mateo pays, set up a linear equation and calculate how much each person must contribute to pay the full amount.

We are designing a new triangular skate park track. The total perimeter of the track must be $54$ meters. The design conditions are as follows:

• The longest side measures $x$ meters.

• The medium side measures half of the longest side plus $3$ meters.

• The shortest side measures $10$ meters less than the longest side.

Find the dimensions of the three sides of the track by solving the corresponding equation.

Marina's age is three times what it was $10$ years ago. If we let $x$ be her current age, can you find out how old she 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 $2$ meters and decrease the length by $4$ meters, the perimeter remains $44$ meters. Calculate the original dimensions of the track.

A games shop sold one third of the consoles it had in stock during the morning. In the afternoon, it sold half of what was left. If there are still $12$ unsold consoles when the shop closes, how many consoles did they start the day with?

Set up the equation by letting $x$ be the total number of consoles at the start of the day.