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) $25 = 4x + 1$

Marta went to the stationer's and bought 3 identical notebooks. If she paid with a $600$ pesos note and received $60$ pesos change, set up an equation where $x$ is the price of one notebook and solve it to find how much each one cost.

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

Marcos 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 $4\,200$ pesos 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 cell phone decreases at a constant rate. During the morning, a quarter of the total charge was used. In the afternoon, a third of what remained after the morning was spent. If $40$% 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 $x$.

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:

Marcos and Lucía live in different villages at a distance of $24 \text{ km}$ apart. At 10:00 am, Lucía sets off towards Marcos' village on her bicycle at a constant speed of $18 \text{ km/h}$. Half an hour later, Marcos sets off to meet Lucía on his electric scooter, travelling at $25 \text{ km/h}$. At what distance from Lucía'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 $12 \text{ liters}$ 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 $\text{RD\$}\ 720$, they have $\text{RD\$}\ 240$ left over for wrapping paper. But if each person contributes $\text{RD\$}\ 600$, they are $\text{RD\$}\ 720$ 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).