First-Degree Equations
At the corner store, one pan of the scale holds 3 identical bags of rice and 2 one-pound weights; the other holds 11 one-pound weights. The scale is balanced. How much does each bag weigh? Without realizing it, you have just set up an equation: $3x + 2 = 11$. Solving it means finding the weight of the bag without unbalancing the scale.
What an equation is
Don't confuse an equation with an identity: $2(x + 1) = 2x + 2$ is true for any $x$, so it is an identity, not an equation. And $3x + 2$ (with no $=$) is just an expression.
Checking a solution is easy: substitute and see whether both sides give the same value. For $x = 3$: $3 \times 3 + 2 = 11$ ✓. For $x = 4$: $3 \times 4 + 2 = 14 \neq 11$ ✗.
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.…
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:
- a) $3x + 5$
- b) $2x - 8 = 10$
- c) $x^2 - 4 = 0$
- d) $5 + 3 = 8$
- 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:…
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:
$5x - (x + 4) = 8$
Solving: the scale
The golden rule: whatever you do on one side, do on the other side too. That way the scale stays balanced and the equality stays true. Solve the scale:
That idea is what we call transposing terms in class:
Solve the following basic equations using transposition of terms (moving terms to the opposite side by changing their operation): a) $x + 15 = 42$ b) $x - 7 = -12$ c)…
Solve the following basic equations using transposition of terms (moving terms to the opposite side by changing their operation):
- a) $x + 15 = 42$
- b) $x - 7 = -12$
- c) $10 - x = 4$
- d) $25 = x + 18$
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…
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.
Parentheses and denominators
Real equations usually come "tangled up." There is an order that always works:
Solve these linear equations by expanding the brackets first. Remember to apply the distributive property carefully: a) $3(x - 2) = 12$ b) $2(x + 4) + 5 = 17$ c)…
Solve these linear equations by expanding the brackets first. Remember to apply the distributive property carefully:
- a) $3(x - 2) = 12$
- b) $2(x + 4) + 5 = 17$
- c) $10 - (x - 3) = 8$
Solve the following linear equation, making sure to simplify all terms and pay attention to sign changes in front of fractions:…
Solve the following linear equation, making sure to simplify all terms and pay attention to sign changes in front of fractions:
$\frac{2(x - 3)}{5} - \frac{x - 1}{2} = \frac{3x - 4}{10} + 1$
Solve the following linear equation with fractional coefficients and nested terms. Pay close attention to handling the negative signs in front of fractions:…
Solve the following linear equation with fractional coefficients and nested terms. Pay close attention to handling the negative signs in front of fractions:
$$\frac{2(x-5)}{3} - \frac{3x - \frac{x+1}{2}}{5} = \frac{x-1}{6} - \frac{7}{10}$$Is there always a solution?
Almost always, but not always. When you simplify, this can happen:
| What you end up with… | 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 (no number times 0 gives 5) |
| $0x = 0$ | $0x = 0$ | Infinitely many solutions: it was an identity |
The drawing previews an idea you will see in functions: each side is a line, and the solution is where they cross.
Given the equation with an unknown coefficient $k$: $$\frac{k \times x}{3} - 4 = \frac{x + 2}{2}$$ Calculate the value that $k$ must have so that the solution to the…
Given the equation with an unknown coefficient $k$:
$$\frac{k \times x}{3} - 4 = \frac{x + 2}{2}$$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.
Irrational coefficients
Coefficients don't have to be whole numbers or fractions: they can be irrational numbers like $\sqrt{2}$ or $\sqrt{3}$. The rules are exactly the same: $\sqrt{2}$ is just another number, treated like any other coefficient.
Linear inequalities
An inequality is like an equation, but with $<$, $>$, $\le$, or $\ge$ instead of $=$. Its solution is not one number, but many: an interval.
It is solved with the same transpositions, with just one new rule:
With $\le$ or $\ge$ the endpoint is included (filled circle and bracket: $(-\infty, -6]$).
With irrational coefficients, same thing: $\sqrt{2}\,x \le 4 \Rightarrow x \le \dfrac{4}{\sqrt{2}} = 2\sqrt{2} \approx 2.83$, that is, $(-\infty, 2\sqrt{2}]$. Since $\sqrt{2} > 0$, the direction does not change.
Setting up word problems
Solving equations is half the work; the other half is translating the problem. Follow these steps:
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Read and decide what the unknown is ($x$ = what they ask you for).
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Express the other quantities using $x$.
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Find the equality the problem states and write the equation.
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Solve and check against the text, not just against the equation.
| Words | Algebra |
|---|---|
| twice a number | $2x$ |
| half of a number | $\dfrac{x}{2}$ |
| 5 more than a number | $x + 5$ |
| three times a number, minus 50 | $3x - 50$ |
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…
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 $x$).
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…
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.
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…
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.
Numbers, digits, and figures
There are three types of problems that keep coming up.
Consecutive numbers: if the first is $x$, the next integers are $x+1$, $x+2$. If they are consecutive even (or odd) numbers, they go up by 2: $x$, $x+2$, $x+4$.
Two-digit numbers: with tens digit $d$ and units digit $u$, the number is worth $10d + u$. Reversed, it is worth $10u + d$.
Figures: use the perimeter or area formula with sides written in terms of $x$.
Find three consecutive even integers such that the sum of the first two is equal to the third plus $14$.
Find three consecutive even integers such that the sum of the first two is equal to the third plus $14$.
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…
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.
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…
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.
Mixtures, prices, and percents
In a mixture, what is conserved is the total amount and the amount of "what matters" (money, acid, salt…). If you take $x$ liters of one kind, you take $(\text{total} - x)$ of the other.
Discounts: if an original price is $x$ and it has a 20% discount, you pay $0.80x$ and the savings are $0.20x$. The percent is always calculated on the original price.
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…
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?
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…
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?
A technology store launches an offer on tablets and accessories. Álvaro buys a tablet and a keyboard for a total of $\text{RD\$}\ 325…
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?
Motion and faucets
Motion. The basic formula is $d = v \times t$. If two vehicles meet, at that moment they are at the same distance from the point you are measuring from. If one leaves earlier, it has traveled for more time.
If they start from points separated by a distance $D$ and the one behind is chasing the other, the one behind must cover $D$ km more: $v_1 t = v_2 t + D$.
Faucets and work. If something fills in $h$ hours, in one hour it fills $\dfrac{1}{h}$ of the pool. You add the parts per hour of whatever helps and subtract those of a leak. If together they take $t$ hours: $t \times (\text{part per hour}) = 1$.
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…
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?
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…
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?
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…
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?
Solving for a letter in formulas
The same rules work for solving for a letter in a formula. From $v = \frac{d}{t}$ (speed) you get $d = v \times t$ and $t = \frac{d}{v}$.
When the letter you want to solve for appears in several terms, gather those terms on one side, take out the letter as a common factor, and divide.
Find the length of side $x$ in the following rectangle, knowing that its total perimetre is $34$ cm.
Find the length of side $x$ in the following rectangle, knowing that its total perimetre is $34$ cm.
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…
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 $a \neq d$:
$$a(x + b) - c = d(x - a)$$Express the final result as a single fraction.
Back to the beginning: each bag weighs 3 pounds, because if you remove 2 pounds from each pan you are left with 3 bags weighing 9 pounds ($3x = 9$), and splitting among the 3 bags gives $x = 3$ pounds.