Exercises: Linear equations

Classify the following mathematical equalities as identities or equations, briefly explaining why in each case:

1. a) $3x + 5 = 20$

1. b) $2(x + 1) = 2x + 2$

1. c) $x + x = 2x$

1. d) $x - 7 = 10$

Look at the following set of scales in balance. If the circular weights represent the unknown $x$ and the square weights represent $1$ kg each, write the equation that represents the image and find the value of $x$ using the logic of balance (removing the same amount from both pans).

Check whether the proposed values for the unknown are solutions to the following equations:

1. a) $2x - 3 = 7$ Is $x = 5$ the solution?

1. b) $\frac{x}{4} + 1 = 6$ Is $x = 20$ the solution?

1. c) $3(x - 2) = 12$ Is $x = 4$ the solution?

Translate these word phrases into algebraic expressions using the letter $x$ to represent the unknown number:

1. a) Double a number plus eight is equal to twenty.

1. b) Half a number minus five gives three.

1. c) A number increased by seven units is equal to three times the number.

1. d) The sum of a number and the next consecutive number is fifteen.

Solve these one-step equations mentally by applying the addition rule (what is added is subtracted on the other side and vice versa):

1. a) $x + 12 = 30$

1. b) $x - 8 = 15$

1. c) $7 + x = 2$

1. d) $x - 10 = -4$

Find the value of $x$ in the following equations by applying the multiplication rule (what is multiplying divides on the other side and vice versa):

1. a) $5x = 45$

1. b) $\frac{x}{3} = 7$

1. c) $-2x = 14$

1. d) $\frac{2x}{5} = 4$

Solve the following simple two-step equations, first collecting the terms with $x$ on one side and the numbers on the other:

1. a) $3x + 4 = 19$

1. b) $5x - 2 = 13$

1. c) $10 - 2x = 4$

1. d) $25 = 4x + 1$

Aoife went to the stationer's and bought 3 identical notebooks. If she paid with a $10$ € note and received $1$ € change, set up an equation where $x$ is the price of one notebook and solve it to find how much each one cost.

Cian has to solve the equation $-3x + 8 = -7$. On his exam, he wrote the following step by step:

• Step 1: $-3x = -7 - 8$

• Step 2: $-3x = -15$

• Step 3: $x = \frac{-15}{3} = -5$

1. a) Point out in which step he made the mistake and explain which rule of transposition of terms (or balance method) he applied incorrectly.
2. b) Correct the exercise by solving the equation properly and check your result numerically in the initial equation to show that it is the correct solution.

Design two different linear equations that satisfy the following conditions:

1. a) An equation with the unknown $x$ on both sides whose solution is exactly $x = -4$. Briefly explain how you constructed it ensuring the balance of the equality.
2. b) An equation of the form $a \times x + b = c \times x + d$ (with $a, b, c, d$ non-zero integers) that has no solution, that is, when simplified it results in an expression of the type $0x = k$ with $k \neq 0$. Justify why no one could find a number that satisfies it.

Solve the following linear equation by correctly applying the distributive property and collecting like terms:

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

Find the value of the unknown $x$ in the following equation involving denominators, first simplifying by finding the lowest common multiple:

$\frac{x - 3}{4} - \frac{x - 5}{6} = \frac{x - 1}{9}$

The perimetre of a rectangular padel court is $56$ metres. We know that the length of the court is $2$ metres more than double its width. Calculate the dimensions of the court by setting up a linear equation.

Cian has bought three strategy games and two battle passes in a digital store. Each battle pass costs half as much as a game. If he paid $70$ € in total, what is the price of a game and what is the price of a battle pass?

A mother's age is three times that of her son. $8$ years ago, the sum of their ages was $48$ years. How old is each of them now?

The battery level of a mobile phone decreases at a constant rate. During the morning, one quarter of the total charge was consumed. In the afternoon, one third of the charge remaining after the morning was used. If $50\,\%$ of the battery still remains ($40$ units of charge), what is the total charge capacity of the phone?

Find three consecutive integers such that the sum of the first and the second is $15$ units greater than the third number.

Look at the following set of scales in balance. If all the blue boxes are identical and weigh $x$ kilograms, and the orange weights show their weight in kg, set up and solve the equation to find the value of $x$.

Look at the following equation with parentheses: $$4 \times \left(\frac{x}{3} - 1\right) = 8$$

Sara and David propose two different strategies to start solving it:

• Sara's method: First apply the distributive property to remove the parentheses by multiplying the 4 by each term inside, and then reduce to a common denominator.

• David's method: Since the 4 is multiplying the entire left-hand side, directly move the 4 dividing to the right-hand side: $\frac{x}{3} - 1 = \frac{8}{4}$.

1. a) Solve the equation following Sara's method step by step.
2. b) Solve the equation following David's method step by step.
3. c) Compare both paths: which one is more direct and reduces the risk of making mistakes with fractions? Would David's method have been valid if the equation had been $4 \times \left(\frac{x}{3} - 1\right) + 2 = 8$? Justify your answer.

In a municipal electric bike rental service in Dublin, the rate for non-subscribed users consists of a fixed fee of $1.20$ € for unlocking plus $0.15$ € for each minute of the ride. At the end of his ride back home, Seán is charged a total of $4.95$ € by the mobile app.

1. a) Write a linear equation where the unknown $m$ represents the duration of the ride in minutes.
2. b) Solve the equation explaining what operation you perform on both sides of the equality at each step to isolate the unknown.
3. c) The service conditions warn that rides exceeding half an hour ($30$ minutes) incur an excess use penalty. Based on your solution, determine whether Seán must pay a penalty or not, indicating how many minutes he had left over or fell short of reaching that limit.

Solve the following linear equation, paying special attention to the negative signs in front of fractions and using the lowest common multiple to simplify denominators:

Cian and Aoife live in different villages at a distance of $24 \text{ km}$ apart. At 10:00 am, Aoife sets off towards Cian's village on her bicycle at a constant speed of $18 \text{ km/h}$. Half an hour later, Cian sets off to meet Aoife on his electric scooter, travelling at $25 \text{ km/h}$. At what distance from Aoife's village will they cross paths, and at what time will they meet?

In a chemistry laboratory, we want to dilute the concentration of a cleaning solution. We have $12 \text{ litres}$ of a liquid that contains $20\%$ pure alcohol. What amount of distilled water (without alcohol) must we add so that the final mixture has an alcohol concentration of only $15\%$?

A graphic design company creates rectangular posters. In its latest model, the base is $10 \text{ cm}$ longer than the height. To comply with a new standard, they decide to increase the base by $5 \text{ cm}$ and decrease the height by $2 \text{ cm}$. If after these changes the perimetre of the new poster is $110 \text{ cm}$, what were the original dimensions of the poster?

A father's current age is three times that of his son. Five years ago, the sum of their ages was equal to the age the father will be in ten years' time. How old are the father and son currently?

A water tank has two inlet taps and a drain. The first tap can fill the tank in $6 \text{ hours}$, and the second tap in $4 \text{ hours}$. If the tank is full, the drain can empty it completely in $8 \text{ hours}$. If we open both inlet taps and the drain at the same time starting with the tank empty, how long will it take to fill completely?

A group of friends wants to buy a birthday present that costs a fixed price. If each person contributes $\text{€}\ 12$, they have $\text{€}\ 4$ left over for wrapping paper. But if each person contributes $\text{€}\ 10$, they are $\text{€}\ 12$ short. How many friends are in the group and how much does the present cost?

Find a three-digit number with the following characteristics: the tens digit is twice the units digit, and the hundreds digit is three times the units digit. If we add the three digits together, the result is the same as subtracting the units digit from the hundreds digit and adding 8. Once you set up the equation, determine what the number is. (Hint: let $x$ be the units digit).

At a Met Éireann mountain weather station, $-2$ ºC is recorded at 8:00 in the morning. Due to the effect of sunlight, the temperature rises at a constant rate of $1.5$ ºC per hour. At the same time, at an observatory located in a nearby moorland, the temperature at 8:00 was $10$ ºC, but the arrival of a cold air mass causes it to drop at a rate of $0.5$ ºC per hour.

1. a) Before calculating anything, reasonably estimate whether the temperatures of both stations will match before or after 13:00 (that is, after 5 hours have passed since 8:00). Justify your estimate by comparing the approximate evolution of both temperatures.
2. b) Set up a linear equation where the unknown $t$ represents the number of hours elapsed since 8:00 needed for both stations to record exactly the same temperature. Solve it step by step.
3. c) Find the exact time of day when they match and the temperature the thermometers will show at that moment. Check whether your estimate from section a) is confirmed and whether the values obtained are consistent.

Consider the following linear equation involving an unknown parameter $k$:

1. a) Apply the distributive property, collect like terms, and write the equation in the reduced form $(A - k) \times x = B$, stating the numerical values of $A$ and $B$.
2. b) Determine what value $k$ must take for the equation to have no solution (case $0x = b$ with $b \neq 0$). Reason why for that specific value it is impossible to find any $x$.
3. c) Is there any possible value of $k$ that makes the equation have infinitely many solutions (identity, case $0x = 0$)? Justify your conclusion by analyzing the two resulting sides of the reduced equation.