Exercises: Linear Equations

Classify the following mathematical equalities into 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 balanced scales. If the circular weights represent the unknown $x$ and the square weights represent $1$ lb each, write the equation that represents the image and find the value of $x$ by applying the logic of balance (removing the same amount from both pans).

Check if 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 sentences from natural language into algebraic language using the letter $x$ to represent the unknown number:

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

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

1. c) A number increased by seven is equal to its triple.

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 moves across as subtracted 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 product rule (what is multiplying moves across as dividing 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 grouping 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) $25 = 4x + 1$

Emma went to the stationery store and bought 3 identical notebooks. If she paid with a $\$10$ bill and received $\$1$ in change, set up an equation where $x$ is the price of one notebook and solve it to find out 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 with denominators, simplifying beforehand by calculating the lowest common multiple:

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

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

Marcus has bought three strategy games and two battle passes in a digital shop. Each battle pass costs half as much as a game. If he has 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 currently?

The battery level of a mobile phone decreases at a constant rate. During the morning, a quarter of the total charge has been consumed. In the afternoon, a third of what was left after the morning has been used. If there is still $40$% battery remaining ($40$ 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 exceeds the third number by $15$ units.

Look at the following balanced scales. If all the blue boxes are identical and weigh $x$ pounds, and the orange weights indicate their weight in lb, 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 the fractions and the use of the lowest common multiple to simplify the denominators:

Marcus and Lucy live in different towns located at a distance of $24 \text{ mi}$. At 10:00 a.m., Lucy leaves for Marcus's town on her bicycle at a constant speed of $18 \text{ mph}$. Half an hour later, Marcus sets off to meet Lucy on his electric scooter, traveling at $25 \text{ mph}$. At what distance from Lucy's town will they cross paths and at what time will the meeting occur?

In a chemistry laboratory, they want to reduce the concentration of a cleaning solution. We have $12 \text{ liters}$ of a liquid containing $20\%$ pure alcohol. What quantity 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 their latest model, the base measures $10 \text{ cm}$ more 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 perimeter 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 the same as the age the father will be in ten years. How old are the father and son currently?

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

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

Find a three-digit number with the following characteristics: the tens digit is twice the ones digit, and the hundreds digit is three times the ones digit. If we add the three digits, the result is the same as if we subtract the ones digit from the hundreds digit and add 10. Once the equation is set up, state what the number is. (Hint: call the ones digit $x$).