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 ones are linear equations. For those that are, state 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 whether a number is a solution to an equation involves substituting the unknown and seeing if the equality holds. Given the following set of values: $\{-2, 0, 3\}$, state with reasoning which one of them is the solution to the equation:

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

Solve the following basic equations using transposition of terms (moving terms to the opposite side by changing their operation):

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 length of side $x$ in the following rectangle, knowing that its total perimetre is $34$ cm.

Solve these linear equations by expanding the brackets first. 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 word problems into algebraic expressions is like learning a new language. Translate and solve the following situation:

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

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

A pair of scales is balanced. On the left pan there are $3$ identical boxes (of weight $x$) and a weight of $2$ kg. On the right pan there is a weight of $14$ kg.

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

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

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

In a tech store, a wireless mouse, a keyboard and headphones cost $6\,600$ pesos in total. We know that the keyboard costs three times as much as the mouse, and that the headphones cost $600$ pesos 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 costing $72$ pesos/liter with mango juice costing $108$ pesos/liter. If they want to get $60$ liters of blend at a price of $84$ pesos/liter, how many liters of pineapple juice should be used?

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

Two cyclists, Lucía and Marcos, take part in a charity race. Lucía sets off from the start line at a constant speed of $20$ km/h. Half an hour later, Marcos sets off from the same point pursuing her at a speed of $25$ km/h. How long will it take Marcos to catch Lucía from the moment 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 handling the negative signs in front of fractions:

A video game designer is configuring the square hitbox of a character. If they increase one of the sides by $4\text{ cm}$ and decrease 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 side length of the original square?

Hint: Although terms in $x^2$ appear, observe how they cancel out when working through.

To fill a village swimming pool, two pipes, A and B, are used. Pipe A can fill the pool by itself 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 turned off. If we turn on pipe A and pipe B simultaneously, the pool fills in exactly $5$ hours (including the leak remaining open). How long would it take pipe B to fill the pool on its own if there were no leak?

In a chemistry laboratory, a student needs to obtain $12$ liters of an acid solution at $25\%$. On the shelf there are only two bottles available: one with a solution of $10\%$ acid and another with a highly concentrated solution at $50\%$. How many liters must be mixed from each bottle to get the exact mixture needed for the experiment?

Two surveillance drones, Helios and Selene, set off from two bases situated at a distance of $150\text{ km}$ from each other. Both fly in the same direction and heading (Helios pursuing Selene). Helios flies at a constant speed of $90\text{ km/h}$ and Selene at $60\text{ km/h}$. 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 base of origin will they meet?

In literal equations, we must be able to rearrange for one variable in terms of the others. Make $x$ the subject of the following equation, 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.

A technology store launches an offer on tablets and accessories. Álvaro buys a tablet and a keyboard for a total of $\text{RD\$}\ 325$. 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 Álvaro saved a total of $\text{RD\$}\ 115$ compared to the original prices, what was the selling price (without discount) of the tablet and of the keyboard?

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