First-Degree Equations
At the corner store you buy three identical notebooks, pay with $\text{RD}\$100$ in cash, and get $\text{RD}\$10$ back. How much does each notebook cost? With a letter and an equation you can answer it. In this section you will learn to solve longer equations, with parentheses and fractions, to set up word problems, and to solve inequalities.
Equation, solution, and identity
To check a solution, substitute the value for the letter and see whether both sides give the same result.
Examples: $x + x = 2x$ is an identity, because it doesn't matter what $x$ is. On the other hand, $x + 3 = 10$ is an equation, because it holds only if $x = 7$.
Classify the following mathematical equalities as identities or equations, briefly explaining why in each case: a) $3x + 5 = 20$ b) $2(x + 1) = 2x + 2$ c) $x + x = 2x$…
Classify the following mathematical equalities as identities or equations, briefly explaining why in each case:
- a) $3x + 5 = 20$
- b) $2(x + 1) = 2x + 2$
- c) $x + x = 2x$
- d) $x - 7 = 10$
Check whether the proposed values for the unknown are solutions to the following equations: a) $2x - 3 = 7$ Is $x = 5$ the solution? b) $\frac{x}{4} + 1 = 6$ Is $x = 20$…
Check whether the proposed values for the unknown are solutions to the following equations:
- a) $2x - 3 = 7$ Is $x = 5$ the solution?
- b) $\frac{x}{4} + 1 = 6$ Is $x = 20$ the solution?
- c) $3(x - 2) = 12$ Is $x = 4$ the solution?
The balance scale and transposition
An equation is like a balance scale in equilibrium: whatever you do to one pan, you do to the other.
That answers the question at the beginning: $3x + 10 = 100$ gives $3x = 90$, so $x = 30$. Each notebook costs $\text{RD}\$30$.
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…
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).
Solve these one-step equations mentally by applying the addition rule (what is added is subtracted on the other side and vice versa): a) $x + 12 = 30$ b) $x - 8 = 15$ c)…
Solve these one-step equations mentally by applying the addition rule (what is added is subtracted on the other side and vice versa):
- a) $x + 12 = 30$
- b) $x - 8 = 15$
- c) $7 + x = 2$
- 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): a) $5x = 45$ b)…
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):
- a) $5x = 45$
- b) $\frac{x}{3} = 7$
- c) $-2x = 14$
- d) $\frac{2x}{5} = 4$
Two-step equations and balance scales with an unknown
When there are numbers and letters on both sides, group the $x$ terms on one side and the numbers on the other. Then solve for $x$.
A balance scale with boxes of $x$ kg on both pans is solved the same way. Write an equation from what you see, remove the same amount from both pans, and solve.
Solve the following simple two-step equations, first collecting the terms with $x$ on one side and the numbers on the other: a) $3x + 4 = 19$ b) $5x - 2 = 13$ c)…
Solve the following simple two-step equations, first collecting the terms with $x$ on one side and the numbers on the other:
- a) $3x + 4 = 19$
- b) $5x - 2 = 13$
- c) $10 - 2x = 4$
- d) $25 = 4x + 1$
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…
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 $x$.
From everyday language to algebraic language
Setting up the problem well is half the work.
| It says… | It is written… |
|---|---|
| a number | $x$ |
| twice a number | $2x$ |
| half of a number | $\dfrac{x}{2}$ |
| a number increased by 7 | $x + 7$ |
| the next number after a number | $x + 1$ |
| three times a number minus 5 | $3x - 5$ |
Translate these word phrases into algebraic expressions using the letter $x$ to represent the unknown number: a) Double a number plus eight is equal to twenty. b) Half a…
Translate these word phrases into algebraic expressions using the letter $x$ to represent the unknown number:
- a) Double a number plus eight is equal to twenty.
- b) Half a number minus five gives three.
- c) A number increased by seven units is equal to three times the number.
- d) The sum of a number and the next consecutive number is fifteen.
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…
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.
Parentheses and denominators together
The usual order is this (if the parentheses are inside the numerators, as in the example, it is easier to remove the denominators first and then the parentheses):
Solve the following linear equation by correctly applying the distributive property and collecting like terms: $4(2x - 3) - 2(5 - 3x) = 3(x + 1) + 5$
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:…
Find the value of the unknown $x$ in the following equation involving denominators, first simplifying by finding the lowest common multiple:
$\frac{x - 3}{4} - \frac{x - 5}{6} = \frac{x - 1}{9}$
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:…
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:
$$\frac{2 \times (x - 3)}{5} - \frac{x - 1}{2} = \frac{1}{4} - \frac{3x - 14}{10}$$Is there always a solution?
When you simplify an equation, there are three possibilities:
| In the end you get… | Example | It means |
|---|---|---|
| $ax = b$ with $a \neq 0$ | $3x = 9$ | One solution: $x = 3$ |
| $0x = b$ with $b \neq 0$ | $0x = 5$ | No solution |
| $0x = 0$ | $0x = 0$ | Infinitely many solutions (identity) |
Word problems with equations: numbers and figures
Method: 1) read and decide what $x$ is; 2) express the other data using $x$; 3) set up the equation; 4) solve; 5) check against the problem statement.
Find three consecutive integers such that the sum of the first and the second is $15$ units greater than the third number.
Find three consecutive integers such that the sum of the first and the second is $15$ units greater than the third number.
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…
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…
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?
Age problems and problems with fractions of a total
In age problems, the trick is a table with the present and the past (or future). Everyone changes by the same amount: $n$ years ago is $-n$ for each person.
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?
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…
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?
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…
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?
Problems about sharing costs, perimeters, and digits
Sharing with a surplus and a shortfall. The total price doesn't change: you write it in two ways and set them equal.
Digits. A two-digit number with tens digit $t$ and units digit $u$ is worth $10t + u$. For a three-digit number, $100h + 10t + u$.
Changes in a figure. First express the new measurements and then use the perimeter formula with them.
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…
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?
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…
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?
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…
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).
Motion, mixture, and faucet problems
Moving objects that meet. If they travel toward each other, the sum of the distances traveled is the total distance. We use $d = v \cdot t$. If one leaves later, its time is shorter.
Mixtures and concentrations. The amount of pure substance doesn't change when you add water: $\text{pure} = \text{percent} \times \text{volume}$.
Faucets and drains. A faucet that fills a tank in $a$ hours does $\dfrac{1}{a}$ of the tank per hour. The ones that fill add up and the drain subtracts. All together must give $1$ tank.
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…
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…
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 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…
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?
First-degree inequalities
A bus fare costs $\text{RD}\$35$ and you have $\text{RD}\$200$. How many rides can you pay for at most? There isn't just one answer, but many: $35x \leq 200$.
The open circle means that $-4$ is not included ($<$ or $>$). With $\leq$ or $\geq$ the circle is filled in and the interval uses a bracket.
Back to the beginning: with $35x \leq 200$ you get $x \leq 5.71$, so you can pay for at most $5$ rides.