Exercises: Linear equation

Look at the following balance scale which is in equilibrium. On the left pan there is a box with an unknown weight $x$ and a $2$ lb weight. On the right pan there is a $7$ lb weight. Write the equation that represents this situation and find the value of $x$ so that the scale remains balanced.

Sophia is organizing her library. If today she bought $12$ new books and now has a total of $45$ books, how many books did she have before the purchase? Set up an equation calling $x$ the initial number of books and solve it.

In a video game shop, all controllers have the same price. If a group of friends has bought $4$ controllers and they have paid a total of $\$120$, what is the price of a single controller? Write the corresponding linear equation and calculate the unknown.

Find the value of $x$ in the following basic equation using the rules for rearranging terms (a term that is added moves to the other side as subtracted and vice versa):

$x + 15 = 32$

Check if the value $x = 4$ is the solution (root) of the following linear equation. To do this, substitute the $x$ with the number $4$ and verify if the equality holds:

$3x - 5 = 7$

The perimeter of a square measures $24$ in. If we call $x$ the length of one of its sides, write the equation that allows you to calculate this side and find its value.

Solve the following linear equation step by step:

$2x + 8 = 18$

Hint: First move the number that does not have $x$ to the other side of the equality by changing its sign.

Translate the following sentence into algebraic language and solve the resulting equation: "Three times a number $x$ is equal to $21$".

In an online streaming service, the price of a standard monthly subscription is $\$x$. The "Premium" subscription costs $\$6$ more than the standard one, and the "Basic" subscription costs half as much as the standard one. If a group of friends pays a total of $\$54$ to purchase one subscription of each type, set up and solve the equation to find the price of the standard subscription.

Find the value of the unknown $x$ in the following equation with parentheses, correctly applying the distributive property and the rules of signs:

$3(2x - 4) - 2(x + 5) = 14$

Solve the following equation with denominators by first simplifying the expression by calculating the least common multiple:

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

Look at the following rectangle. Knowing that its perimeter is $34\text{ cm}$, set up a linear equation and calculate the lengths of its sides and its area.

A mother's age is three times that of her son. $8$ years ago, the sum of their ages was $44$ years. What is the current age of each one? Organize the data in a table before setting up the equation.

Three siblings share $\$130$. The eldest receives twice as much as the middle one, and the latter receives $\$10$ more than the youngest. Set up an equation where $x$ is the amount the youngest sibling receives and find out how much money each one receives.

Solve the following linear equation, making sure to correctly operate the negative signs in front of the fractions:

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

In a sporting goods store, the price of a soccer jersey is $\$x$. During the sale, a discount of $\$12$ is applied. A team buys $8$ discounted jerseys for a total of $\$384$. What was the original price of the jersey before the discount?

Solve the following linear equation step by step, simplifying the denominators using the least common multiple and carefully expanding the parentheses:

$\frac{3(x - 2)}{4} - \frac{2(x + 3)}{3} = \frac{x - 1}{12} - \frac{5x - 4}{2}$

A sustainable courier company uses electric scooters for its deliveries. The monthly maintenance cost of a scooter is $45\text{ \$}$, plus $0,15\text{ \$}$ for each mile travelled. Last month, the total cost of one scooter was equal to that of another model whose fixed maintenance is $60\text{ \$}$, but only costs $0,10\text{ \$}$ per mile. Calculate how many miles both scooters travelled for the expense to be the same.

Find a number such that the sum of its half, its third and its quarter exceeds by $15$ units twice the difference between this number and $10$. Set up the corresponding equation and solve it.

Calculate the value of $x$ so that the perimeter of the following composite shape (a rectangle with a square cut-out in one corner) is $84\text{ cm}$. Note that all measurements are expressed in terms of $x$.

In an electronics store, a user buys three types of peripherals: mice, keyboards and headphones. The price of a keyboard is twice that of a mouse, and the headphones are $15\text{ \$}$ more expensive than the keyboard. If they buy $3$ mice, $2$ keyboards and $1$ pair of headphones, and the total bill with an applied discount of $10\text{ \$}$ is $175\text{ \$}$, what is the price of each item?

Find three consecutive integers such that the difference between the square of the largest and the square of the smallest is equal to $96$.

Hint: Remember that $(x+1)^2 = x^2 + 2x + 1$. Although squares appear, you will see that the resulting equation is linear.

Two water tanks have different capacities. Tank $A$ contains three times as many gallons as tank $B$. If we extract $20$ gallons from tank $A$ and add $10$ gallons to tank $B$, tank $A$ will still have twice as much water as $B$. How many gallons were initially in each tank?

Solve the following equation containing nested parentheses:

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