Exercises: Linear equations

Solve the following linear equation by first expanding the brackets:

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

Calculate the value of $x$ in the following equation with denominators:

$\frac{2x - 1}{3} + \frac{x + 2}{4} = 5$

Find the solution to the equation by putting the terms over a common denominator:

$\frac{x}{2} - \frac{3(x - 1)}{5} = \frac{x + 4}{10} - 1$

Two equations are equivalent if they have the same solution. Check whether the equations $3x - 5 = 7$ and $2x + k = 10$ are equivalent when $k = 2$.

Without solving it completely, expand and state how many real solutions the following linear equation has:

$3(2x - 1) + 4 = 6x + 1$

Three times a number minus half of the same number equals $45$. Set up a linear equation and find the number.

The perimeter of a rectangular plot is $48\text{ m}$. Knowing that the base is exactly twice the height, set up an equation to find the lengths of the sides of the plot.

Carlos is three times as old as his sister Ana. In $5\text{ years}$' time, the sum of their ages will be $34\text{ years}$. Set up a linear equation to find out their current ages.

Marcos has tried to solve the following linear equation in his notebook:

His working out was as follows:

• Step 1: $5 - 2x - 6 = 19$

• Step 2: $-1 - 2x = 19$

• Step 3: $-2x = 20$

• Step 4: $x = \frac{20}{2} = 10$

1. a) Identify the two mistakes made by Marcos (one in step 1 and another in step 4) and explain why they are incorrect based on the distributive property and the balance method.
2. b) Solve the equation correctly step by step and check the obtained result by substituting it into the initial expression.

In a Spanish city, the official taxi fare sets a fixed base fare (flag drop) of $2,50$ € and an additional cost of $1,20$ € for each kilometre travelled. Lucía takes a taxi from the train station to her home, and at the end of the trip, the taximeter displays a total amount of $13,30$ €.

1. a) State a linear equation that models the situation, clearly stating what the unknown $x$ represents.
2. b) Solve the equation, explaining each transposition of terms you carry out to calculate how many kilometres she has travelled.
3. c) Check numerically in the context of the problem that the distance obtained produces exactly the stated total cost.

Solve the following linear equation involving fractions and nested brackets:

$\frac{3(x - 2)}{4} - \frac{2(x + 1)}{6} = \frac{x}{3} - \frac{1}{2}$

Find the solution to the equation by working with decimals:

$0,5(2x - 4) + 1,2(x + 3) = 3,4x - 0,8$

In a coffee shop, two types of coffee beans are mixed. The first costs $6\text{ \euro/kg}$ and the second $8\text{ \euro/kg}$. If $50\text{ kg}$ of mixture is required at a final price of $6,60\text{ \euro/kg}$, set up an equation to find out how many kilograms of each type of coffee should be used.

Two freight trains leave two stations A and B separated by a distance of $450\text{ km}$ at the same time, travelling on parallel tracks in opposite directions. The train leaving A travels at $120\text{ km/h}$ and the one leaving B at $80\text{ km/h}$. Set up an equation to determine how long it will take for them to pass each other and at what distance from station A they will meet.

A theatre sells stalls tickets for $25\text{ \euro}$ and circle tickets for $15\text{ \euro}$. For Saturday's performance, a total of $320\text{ entradas}$ were sold, taking $6300\text{ \euro}$ in receipts. Set up a linear equation to calculate how many tickets of each type were sold.

Calculate the value that the parameter $m$ must take so that the solution to the following equation is $x = 4$:

$\frac{mx - 2}{5} - \frac{x - 1}{3} = x - 3$

There is a rectangle whose base is $5\text{ cm}$ greater than its height. If the base is increased by $2\text{ cm}$ and the height is decreased by $1\text{ cm}$, the perimeter of the new rectangle is $36\text{ cm}$. Set up an equation to calculate the dimensions of the original rectangle.

Tap A fills an empty swimming pool in $6\text{ hours}$, and tap B fills it in $4\text{ hours}$. If both taps are opened at the same time, set up a linear equation to find out how long they will take to fill the pool together.

To solve the equation $\frac{3x - 1}{4} - \frac{x + 2}{6} = \frac{7}{12}$, Sara and David propose two different strategies:

• Sara proposes Method A: calculate the LCM of the denominators ($12$), multiply both sides by $12$ from the start to clear denominators, and work only with integers.

• David proposes Method B: split each fraction by separating the unknown from the constant terms, writing $\frac{3}{4}x - \frac{1}{4} - \frac{1}{6}x - \frac{2}{6} = \frac{7}{12}$, and work by grouping fractions.

1. a) Solve the equation using both methods, checking that they lead to the same solution.
2. b) Justify which of the two methods you consider safer to avoid calculation errors, pointing out in which step of David's method it is most common to make a sign error.

The two-monthly water bill for a household in Seville consists of a fixed service fee of $14,60$ € plus a cost of $1,80$ € per cubic metre ($m^3$) of water consumed. In the last bill received, the total amount payable is $59,60$ €.

1. a) Estimate: Without performing exact calculations, reason mentally whether the two-monthly consumption will have been closer to $10\text{ m}^3$, $25\text{ m}^3$, or $50\text{ m}^3$. Justify your estimate.
2. b) Set up and solve: Define the unknown $x$, write the corresponding linear equation, and solve it step by step to find the exact volume of water consumed.
3. c) Assess: Compare the result with your estimate. Knowing that the average consumption in Spain is $130$ litres per person per day (INE data) and that $2$ people live in the house during the $60$ days of the two-month period (which would represent about $15,6\text{ m}^3$ in total), assess whether the consumption obtained reflects normal usage or if the family should check the installation for a potential leak.

Solve the following equation by first expanding the algebraic identities. Notice how the quadratic terms cancel out upon simplifying:

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

Given the literal equation $a(x - 2) + 3 = 4x - b$, discuss the number of solutions the equation will have depending on the values of the parameters $a$ and $b$.

A cyclist sets off from a city at a constant speed of $15\text{ km/h}$. Exactly one hour later, a car sets off in pursuit from the same point at $90\text{ km/h}$. Set up an equation to calculate how long it will take the car to catch up with the cyclist and at what distance from the city they will meet.

A jeweller has a gold ingot with $80\%$ purity and a second ingot with $95\%$ purity. Set up an equation to find out how much of each ingot should be melted down to obtain a new ingot of exactly $3\text{ kg}$ with $89\%$ purity.

In a multiple-choice exam consisting of $40\text{ questions}$, $3\text{ points}$ is awarded for each correct answer, $1\text{ point}$ is deducted for each incorrect answer, and unanswered questions neither add nor subtract points. A student left $4\text{ questions}$ unanswered and obtained a final score of $60\text{ points}$. Set up a linear equation to calculate how many questions they answered correctly.

Calculate the value of $x$ in the following literal equation, assuming that $a \neq 0$, $b \neq 0$ and $a \neq b$:

$\frac{x - a}{b} + \frac{x - b}{a} = 2$

An electronics shop applies a discount of $15\%$ to the retail price of a laptop. At the checkout, $21\%$ VAT is added to the already discounted price. If the customer pays $822,80\text{ \euro}$ in the end, set up a linear equation to calculate the initial price of the laptop before the discount and before VAT.

The sum of the ages of three siblings is $44\text{ years}$. The middle sibling is $3\text{ years}$ older than the youngest sibling, and the age of the oldest sibling is equal to the sum of the ages of the two younger siblings minus $2\text{ years}$. Set up a linear equation to determine the exact age of each of the three siblings.

A mathematics teacher wants to design an exam exercise with the structure $\frac{2x - a}{3} - \frac{x - 1}{4} = b$, where $a$ and $b$ must be integers chosen by him.

1. a) If the teacher needs the solution of the equation to strictly be $x = 5$, find a pair of integers $(a, b)$ that satisfies that condition.
2. b) Is there a single possible pair $(a, b)$ or are there more? Deduce a mathematical relationship between $a$ and $b$ that always guarantees that $x = 5$ is a solution, and provide a second pair of integer values different from those in the previous section.
3. c) Would it be possible to choose values for $a$ and $b$ so that the resulting equation has no solution (inconsistent equation)? Justify your answer by analysing the coefficients of the variable $x$.

A recent vocational training (FP) graduate in Spain receives two job offers as a solar panel installation and maintenance technician:

• Company North: Offers a fixed base salary of $1100$ € per month plus a commission of $45$ € for each completed installation.

• Company South: Offers a fixed base salary of $1340$ € per month plus a commission of $25$ € for each completed installation.

1. a) Model the situation by setting up and solving a linear equation to determine with how many monthly installations ($x$) the gross monthly salary is exactly the same at both companies.
2. b) According to industry statistics, a novice professional completes an average of $8$ installations per month, whereas an experienced one reaches $16$. Based on your equation, make a reasoned argument about which company an entry-level technician should choose and which one an experienced technician should choose.
3. c) If a fixed personal income tax (IRPF) withholding of $12\,\%$ is applied to both salaries, write the new equation that equates the net salaries and justify mathematically, without needing to solve it again, why the break-even number of installations does not change.