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 balance scale in perfect equilibrium. Look at the following image where the orange circles represent our unknown $x$ and the grey squares represent $1$ kg units.

Write the equation representing the balance of the scale:

In your favourite 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) $2x - 1 = 5$

Aisha has a collection of football 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 Aisha had, write the equation representing this problem and solve it to find out how many cards she had at the start.

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) $\frac{x}{3} = 5$

The perimeter of a rectangular plot measures $60$ metres. We know that the length of the plot is $20$ metres. 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 down the steps to solve for $x$ and calculate the width of the plot.

Three friends went to have a snack and ordered three identical milkshakes. When paying with a $20$ € note, they received $11$ € 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: $3 \cdot (\text{precio batido}) + \text{vuelta} = \text{total pagado}$)

Marcos tried to solve the equation $4x - 6 = 14$ in his notebook and wrote the following step-by-step resolution:

Step 1: $4x = 14 - 6$

Step 2: $4x = 8$

Step 3: $x = \dfrac{8}{4} = 2$

1. a) Identify the mistake Marcos made when going from the original statement to Step 1 and explain which basic rule of term transposition he did not follow.
2. b) Solve the equation correctly, showing all the steps.
3. c) Check your solution by substituting the value obtained into the initial equation to show that the equality holds.

Imagine you are preparing a test for your classmates and make up a linear equation that simultaneously satisfies these three conditions:

1. Its exact solution must be $x = -2$.

2. The unknown $x$ must necessarily appear on both sides of the equation.

3. At least one of the coefficients multiplying $x$ must be greater than $1$.

Clearly explain the reasoning you followed to construct your equation and prove that it is correct by checking that, when substituting $x = -2$, both sides give the same numerical value.

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.

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

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

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.

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

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?

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.

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

At an AEMET weather station in Córdoba, during a spring day, the temperature at 9:00 was $14\ ^\circ\text{C}$. From that time on, the temperature increased at a constant rate of $3\ ^\circ\text{C}$ for each passing hour until reaching the maximum temperature of the day, which was $29\ ^\circ\text{C}$.

1. a) Before setting up any algebraic operation, make a mental estimate: do you think the maximum temperature was reached before 13:00 or after? Briefly justify your prediction.
2. b) Let $t$ be the number of hours that have passed since 9:00. Set up a linear equation that models the situation and solve it step by step to find out how many hours passed.
3. c) Determine at what exact time the $29\ ^\circ\text{C}$ was reached and assess whether the numerical result confirms your previous estimate and is consistent with the daily pattern.

To travel daily to high school and to his basketball practice, Pablo analyses two fares for the metropolitan transport card:

• Youth Fare (Tarifa Joven): a fixed monthly pass of $15$ euros plus $0,60$ euros per trip taken.

• Pay-as-you-go Fare (Tarifa Monedero): no fixed monthly fee, but each trip costs $1,10$ euros.

1. a) Set up a linear equation that allows calculating the number of trips $v$ for which the monthly cost would be identical in both options.
2. b) Solve the equation to find that number of trips.
3. c) If Pablo estimates that he will take approximately $36$ trips per month, interpret the result obtained in the previous section to advise him on which of the two fares will be more economical for him. Justify your advice by comparing the expected costs.

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:

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

Three friends decide to buy an annual subscription to a streaming platform that costs a total of $165$ €. Based on how much they use it, they have agreed that:

• Sara will pay twice as much as Lucas.

• Lucas will pay $15$ € 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 $54$ metres. The design conditions are as follows:

• The longest side measures $x$ metres.

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

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

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

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.

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 of the new rectangle becomes $44$ meters. Find the original dimensions of the track.

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.

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.

Given the equation with parentheses and fractions:

$\dfrac{3(x - 1)}{2} - \dfrac{2x + 4}{5} = 2$

Two students propose different solving strategies:

• Elena's strategy: First multiply all the terms on both sides by the least common multiple of the denominators ($10$) to eliminate the fractions in one go, and then remove the parentheses.

• David's strategy: First apply the distributive property to the numerators, split each fraction into simple fractions, and operate by adding the terms with $x$ and the numerical terms using fraction arithmetic.

1. a) Solve the equation in an orderly manner following Elena's strategy.
2. b) Solve the equation following David's strategy.
3. c) Compare both paths: which one do you consider more effective for avoiding common sign errors and simplifying intermediate calculations in 1st year of ESO? Justify your answer by analysing a critical point where it is easy to make a mistake in either of the two methods.

A town hall wants to enclose a rectangular school garden using a donated batch of wire fencing with a total length of $68\text{ metres}$, which must be fully used to fence the entire perimeter.

1. a) The technical project sets as a condition that the length of the garden must measure $4\text{ metres}$ more than three times its width. Let $x$ be the width in metres, write the algebraic expression for the length, and formulate the equation representing the total perimeter.
2. b) Solve the equation and calculate the exact dimensions (width and length) of the planned garden.
3. c) Suppose the neighbourhood association asks to modify the specifications: they want the length to be four times the width, using exactly the same $68\text{ metres}$ of fencing. Set up the new equation and solve it. Based on the value of $x$ obtained, reflect: is it possible to build the garden under these new conditions if the boundary stakes can only be placed at distances that are whole numbers of metres? Justify your conclusion.