Solving Linear Equations
At the science museum there is a balance scale. On one pan sit 3 mystery boxes and two 1-lb weights. On the other pan sit eleven 1-lb weights. The scale is perfectly level. How much does each box weigh? Without noticing, you just wrote an equation: $3x + 2 = 11$. Solving it means finding the weight of a box without tipping the scale.
What Is an Equation?
A linear equation is one where the variable appears only to the first power (no $x^2$, no $\frac{1}{x}$). Here is a picture of the parts:
Don't mix up three things:
| It looks like… | Example | What it is |
|---|---|---|
| no $=$ sign | $3x + 2$ | an expression (you can simplify it, not solve it) |
| true only for some values | $3x + 2 = 11$ | an equation |
| true for every value | $2(x + 1) = 2x + 2$ | an identity |
Checking a solution is easy: substitute the number for the variable and see whether both sides match. Is $x = 5$ a solution of $4(x - 1) = 2x + 6$? Left side: $4(5 - 1) = 16$. Right side: $2(5) + 6 = 16$. They match, so yes ✓. Try $x = 3$: the left side is $8$ and the right side is $12$, so no ✗.
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…
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:
- a) $3x + 5$
- b) $2x - 8 = 10$
- c) $x^2 - 4 = 0$
- d) $5 + 3 = 8$
- 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:…
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:
$5x - (x + 4) = 8$
Solving with the Balance Idea
The golden rule: whatever you do to one side, do to the other side. The scale stays level, and the equation stays true. Solve the balance below by removing the same weight from both pans, then splitting what is left evenly:
To solve, you use inverse operations to undo whatever was done to the variable. Undo in the reverse order of the order of operations: first undo addition and subtraction, then undo multiplication and division.
Some problems call this shortcut transposition of terms ("moving a term to the other side and changing its operation"). Underneath, you are always subtracting from, adding to, multiplying, or dividing both sides, so think of it that way.
Solve the following basic equations using the transposition of terms (what is adding moves over as subtracting and vice versa): a) $x + 15 = 42$ b) $x - 7 = -12$ c)…
Solve the following basic equations using the transposition of terms (what is adding moves over as subtracting and vice versa):
- a) $x + 15 = 42$
- b) $x - 7 = -12$
- c) $10 - x = 4$
- d) $25 = x + 18$
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…
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.
Parentheses, Like Terms, and Variables on Both Sides
Real equations come tangled up. This order of steps always works:
How many solutions can there be?
Almost always one, but not always. After you simplify, one of three things happens:
| You end up with… | Example | Meaning |
|---|---|---|
| $ax = b$ with $a \neq 0$ | $3x = 9$ | One solution: $x = 3$ |
| a false statement | $0 = 5$ | No solution |
| a true statement | $0 = 0$ | Infinitely many solutions (an identity) |
Think of each side of the equation as a line. The solution is where the two lines meet. You will see much more about this soon.
When $x^2$ cancels
Sometimes you multiply two binomials and get $x^2$ on both sides. Don't panic: the $x^2$ terms cancel and a linear equation remains.
Solve these linear equations by first removing the parentheses. Remember to apply the distributive property carefully: a) $3(x - 2) = 12$ b) $2(x + 4) + 5 = 17$ c)…
Solve these linear equations by first removing the parentheses. Remember to apply the distributive property carefully:
- a) $3(x - 2) = 12$
- b) $2(x + 4) + 5 = 17$
- c) $10 - (x - 3) = 8$
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 $2\text{ cm}$,…
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.
Equations with Fractions
Fractions make equations look scary, but there is a trick: multiply every term by the LCD (the least common multiple of the denominators). All the denominators vanish.
Solve the following linear equation, making sure to simplify all terms and paying attention to sign changes in front of the fractions: $$\frac{2(x - 3)}{5} - \frac{x -…
Solve the following linear equation, making sure to simplify all terms and paying attention to sign changes in front of the fractions:
$$\frac{2(x - 3)}{5} - \frac{x - 1}{2} = \frac{3x - 4}{10} + 1$$Solve the following linear equation with fractional coefficients and nested terms. Pay close attention to the management of negative signs in front of the fractions:…
Solve the following linear equation with fractional coefficients and nested terms. Pay close attention to the management of negative signs in front of the fractions:
$$\frac{2(x-5)}{3} - \frac{3x - \frac{x+1}{2}}{5} = \frac{x-1}{6} - \frac{7}{10}$$Solving for a Letter: Literal Equations
The same rules let you solve for one letter in a formula or in an equation with several letters (a literal equation). From $d = rt$ you get $t = \dfrac{d}{r}$. The other letters are treated like numbers.
When the letter you want appears in several terms, collect those terms on one side and factor out the letter.
The same skills help when an equation has an unknown coefficient. If you are told what $x$ is, plug it in and solve for the other letter.
Given the equation with an unknown coefficient $k$: $$\frac{k \cdot 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 \cdot x}{3} - 4 = \frac{x + 2}{2}$$Calculate the value that $k$ must have so that the solution to the equation is $x = 6$. Once $k$ is found, check that the equality holds by substituting both values into the original equation.
In literal equations, we must be able to solve for a variable in terms of the others. Solve for the variable $x$ in the following equality, assuming that all parameters…
In literal equations, we must be able to solve for a variable in terms of the others. Solve for the variable $x$ in the following equality, assuming that all parameters $a, b, c$ and $d$ are known and non-zero, with $a \neq d$:
$$a(x + b) - c = d(x - a)$$Express the final result as a single fraction.
Translating Words into Equations
Word problems are like learning a new language: you translate English into algebra. Here is a mini dictionary:
| In words | In algebra |
|---|---|
| a number | $x$ |
| the sum of a number and 5, 5 more than a number | $x + 5$ |
| 5 less than a number | $x - 5$ |
| 5 subtracted from a number | $x - 5$ |
| twice a number, double a number | $2x$ |
| half of a number | $\dfrac{x}{2}$ |
| is, equals, gives, results in | $=$ |
Perimeter problems come with a formula. For a rectangle with length $\ell$ and width $w$, the perimeter is $P = 2\ell + 2w$.
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…
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$).
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.
Find the value of the side $x$ in the following rectangle knowing that its total perimeter is $34$ in.
Find the value of the side $x$ in the following rectangle knowing that its total perimeter is $34$ in.
Number and Price Problems
Many problems describe several unknowns that are all related to one of them. Pick the simplest one as $x$ and write the others in terms of $x$.
Consecutive integers differ by 1: $x,\ x + 1,\ x + 2$. Consecutive even integers (or consecutive odd integers) differ by 2: $x,\ x + 2,\ x + 4$.
Digit problems. If a two-digit number has tens digit $t$ and units digit $u$, its value is $10t + u$. With the digits reversed, it is $10u + t$.
In an electronics store, a wireless mouse, a keyboard, and headphones cost a total of $\$110…
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.
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…
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$.
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…
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.
Mixtures, Discounts, and Percents
Mixtures. The key idea: the amount of the "ingredient" (money, acid, juice concentrate) in the mix equals the sum of the amounts in the parts.
$$\text{amount} \times \text{rate} \;(\text{price or percent}) = \text{total of that ingredient}$$Prices with cents, like $\$1.20$ per gallon, work exactly the same way: write them as decimals and multiply by the number of gallons.
Acid problems work the same way. A $10\%$ solution has $0.10$ gallons of acid in every gallon, so $x$ gallons of it contain $0.10x$ gallons of pure acid.
Discounts. A $20\%$ discount means you pay $80\%$ of the original price, so the sale price is $0.8x$, and the amount you save is $0.2x$.
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…
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?
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 $10\%$…
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?
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…
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?
Rate Problems: Motion and Work
Motion
The key formula is $d = rt$: distance = rate × time. When one traveler catches up with another, both are the same distance from the starting point at that moment, so you set the two distance expressions equal.
The graph shows distance against time (hours since Maya left). The lines cross at the catch-up point: $1.5$ hours and $4.5$ miles. Maya's line is $d = 3t$ and her brother's is $d = 9(t - 1)$.
A gap instead of a head start. If the slower traveler starts $g$ miles ahead, the faster one must travel $g$ miles more: faster's distance = slower's distance + $g$.
Work and filling problems
Think in rates: a pump that fills a tank in 10 hours does $\frac{1}{10}$ of the job per hour. Rates working together add. Something that works against you, like a leak, is a negative rate. In $t$ hours, the fraction done is (rate) × $t$, and a finished job counts as $1$.
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…
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?
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…
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?
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…
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?
Back to the start: each mystery box weighs 3 lb. Take 2 lb off both pans and 3 boxes balance 9 lb ($3x = 9$). Split the 9 lb among 3 boxes, and $x = 3$ lb.