Exercises: Linear equations

In mathematics, it is essential to distinguish between an algebraic expression and an equation. Look at the following list and identify which are linear equations. For those that are, indicate their left-hand side and their unknown:

1. a) $3x + 5$
2. b) $2x - 8 = 10$
3. c) $x^2 - 4 = 0$
4. d) $5 + 3 = 8$
5. e) $\frac{y}{2} + 1 = 7$

Checking if a number is a solution to an equation consists of substituting the unknown and seeing if the equality holds. Given the following set of values: $\{-2, 0, 3\}$, indicate with reasoning which of them is the solution to the equation:

$5x - (x + 4) = 8$

Solve the following basic equations using the transposition of terms (what is adding moves over as subtracting and vice versa):

1. a) $x + 15 = 42$
2. b) $x - 7 = -12$
3. c) $10 - x = 4$
4. d) $25 = x + 18$

In video game development, equations are sometimes used to calculate scores. A player has three times as many points as their friend. If we subtract $50$ points from the player, they now have exactly $250$ points.

Set up an equation where $x$ is the friend's score and solve it to find how many points each one has.

Find the value of the side $x$ in the following rectangle knowing that its total perimeter is $34$ in.

Solve these linear equations by first removing the parentheses. Remember to apply the distributive property carefully:

1. a) $3(x - 2) = 12$
2. b) $2(x + 4) + 5 = 17$
3. c) $10 - (x - 3) = 8$

Translating statements from natural language to algebraic language is like learning a new language. Translate and solve the following situation:

"If you add half of a number to double the number, you get 25".

(Hint: call the unknown number $x$).

A balance scale is in equilibrium. On the left pan, there are $3$ identical boxes (of weight $x$) and a $2$ lb weight. On the right pan, there is a $14$ lb weight.

Set up the equation that represents this equilibrium and calculate how much each box weighs.

Solve the following linear equation, making sure to simplify all terms and paying attention to sign changes in front of the fractions:

In an electronics store, a wireless mouse, a keyboard, and headphones cost a total of $\$110$. We know that the keyboard costs three times as much as the mouse, and that the headphones cost $\$10$ more than the keyboard. Set up a linear equation and calculate the price of each item.

A juice manufacturer wants to create a new "Tropical" blend by combining pineapple juice that costs $\$1,20$/gallon with mango juice that costs $\$1,80$/gallon. If they want to obtain $60$ gallons of the blend at a price of $\$1,40$/gallon, how many gallons of pineapple juice must they use?

Find three consecutive even integers such that the sum of the first two is equal to the third plus $14$ units.

Hint: If the first even number is $2x$, the next one is $2x + 2$.

Two cyclists, Lucy and Marcus, participate in a charity race. Lucy leaves the starting line at a constant speed of $20$ mph. Half an hour later, Marcus leaves from the same point chasing her at a speed of $25$ mph. How long will it take Marcus to catch up with Lucy from the time he leaves? At what distance from the starting point will they meet?

Given the equation with an unknown coefficient $k$:

Calculate the value that $k$ must have so that the solution to the equation is $x = 6$. Once $k$ is found, check that the equality holds by substituting both values into the original equation.

Solve the following linear equation with fractional coefficients and nested terms. Pay close attention to the management of negative signs in front of the fractions:

A video game designer is configuring the square collision area of a character. If they increase one of the sides by $4\text{ cm}$ and reduce the other by $2\text{ cm}$, the area of the resulting rectangle is $20\text{ cm}^2$ greater than the area of the original square. What is the measure of the side of the original square?

Hint: Although terms in $x^2$ appear, observe how they simplify when calculating.

To fill a town's municipal swimming pool, two pipes, A and B, are used. Pipe A is capable of filling the pool on its own in $10$ hours. Due to a fault in the drainage system, there is a leak at the bottom that would empty the full pool in $30$ hours if the pipes were closed. If we open pipe A and pipe B simultaneously, the pool fills in exactly $5$ hours (accounting for the open leak). How long would pipe B take to fill the pool on its own if there were no leak?

In a chemistry laboratory, a student needs to obtain $12$ gallons of an acid solution at $25\%$. On the shelf, they only have two bottles: one with a solution at $10\%$ acid and another with a highly concentrated solution at $50\%$. How many gallons must they mix from each bottle to obtain the exact mixture needed for their experiment?

Two surveillance drones, Helios and Selene, depart from two bases located at a distance of $150\text{ mi}$ from each other. Both fly in the same direction (Helios chasing Selene). Helios flies at a constant speed of $90\text{ mph}$ and Selene at $60\text{ mph}$. If both take off at the same time, how long will it take Helios to catch up with Selene and at what distance from its home base will the encounter occur?

In literal equations, we must be able to solve for a variable in terms of the others. Solve for the variable $x$ in the following equality, assuming that all parameters $a, b, c$ and $d$ are known and non-zero, with $a \neq d$:

Express the final result as a single fraction.

An electronics store launches an offer on tablets and accessories. Alex buys a tablet and a keyboard for a total of $\$325\text{ \$}$. He knows that the tablet has a $20\%$ discount applied to its original price, while the keyboard, being an older model, has a $50\%$ discount. If Alex has saved a total of $\$115\text{ \$}$ compared to the original prices, what was the selling price (without discount) of the tablet and the keyboard?

Find a two-digit number knowing that the units digit exceeds the tens digit by $3$ units. If we add to this number the number that results from reversing the order of its digits, we obtain $121$ as a result. Set up a linear equation with a single unknown to find the original number.