Solving Linear Equations
At a food truck, 3 tacos and a $\$1$ drink cost $\$10$ in total, and every taco costs the same. How much is one taco? You can probably guess the answer, but guessing stops working when the numbers get messy. In this lesson you'll learn a method that always works, from simple equations all the way to ones with parentheses, fractions, and variables on both sides.
Equations and Solutions
For the food truck, if $x$ is the price of one taco in dollars, the equation is $3x + 1 = 10$. Try different values for $x$ on the scale below and watch when it balances.
To check a solution, substitute the value for the variable and see whether both sides give the same number.
Check if 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…
Check if 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?
Balance Scales
An equation is like a balanced scale. The two sides weigh the same, and they keep weighing the same as long as you do the same thing to both sides. That one rule solves every equation in this lesson.
When you write the steps with symbols, "taking the same thing off both pans" becomes "subtract the same number (or term) from both sides."
Look at the following balanced scales. If the circular weights represent the unknown $x$ and the square weights represent $1$ lb each, write the equation that represents…
Look at the following balanced scales. If the circular weights represent the unknown $x$ and the square weights represent $1$ lb each, write the equation that represents the image and find the value of $x$ by applying the logic of balance (removing the same amount from both pans).
Look at the following balanced scales. If all the blue boxes are identical and weigh $x$ pounds, and the orange weights indicate their weight in lb, set up and solve the…
Look at the following balanced scales. If all the blue boxes are identical and weigh $x$ pounds, and the orange weights indicate their weight in lb, set up and solve the equation to find the value of $x$.
One-Step Equations
To solve an equation, undo what was done to the variable using the inverse operation, on both sides. Some problems say a term "moves across" the equal sign with the opposite operation. That is just shorthand for doing the inverse operation to both sides.
| Equation | What happened to $x$ | Undo it by |
|---|---|---|
| $x + 9 = 25$ | $9$ was added | subtracting $9$ |
| $x - 6 = 11$ | $6$ was subtracted | adding $6$ |
| $7x = 56$ | multiplied by $7$ | dividing by $7$ |
| $\dfrac{x}{4} = 5$ | divided by $4$ | multiplying by $4$ |
Solve these one-step equations mentally by applying the addition rule (what is added moves across as subtracted 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 moves across as subtracted 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 product rule (what is multiplying moves across as dividing and vice versa): a) $5x = 45$ b)…
Find the value of $x$ in the following equations by applying the product rule (what is multiplying moves across as dividing and vice versa):
- a) $5x = 45$
- b) $\frac{x}{3} = 7$
- c) $-2x = 14$
- d) $\frac{2x}{5} = 4$
Two-Step Equations
In $3x + 1 = 10$, two things happened to $x$: it was multiplied by 3, then 1 was added. To undo them, work in reverse order: first undo the adding or subtracting, then undo the multiplying or dividing. It's like unwrapping a gift: the last layer put on is the first one off.
Solve the following simple two-step equations, first grouping 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 grouping 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) $25 = 4x + 1$
Emma went to the stationery store and bought 3 identical notebooks. If she paid with a $\$10$ bill and received $\$1$ in change, set up an equation where $x$ is the…
Emma went to the stationery store and bought 3 identical notebooks. If she paid with a $\$10$ bill and received $\$1$ in change, set up an equation where $x$ is the price of one notebook and solve it to find out how much each one cost.
Translating Words into Equations
Before you can solve a word problem, you need to translate English into algebra. Choose a letter for the unknown number, then turn each phrase into an expression.
| Words | Algebra |
|---|---|
| a number increased by 7; 7 more than a number | $x + 7$ |
| 5 less than a number; a number decreased by 5 | $x - 5$ |
| double (twice) a number; triple a number | $2x$; $3x$ |
| half of a number | $\dfrac{x}{2}$ |
| the next consecutive integer | $x + 1$ |
| is, equals, gives, is the same as | $=$ |
Translate these sentences from natural language into algebraic language using the letter $x$ to represent the unknown number: a) Double a number plus eight is equal to…
Translate these sentences from natural language into algebraic language using the letter $x$ to represent the unknown number:
- a) Double a number plus eight is equal to twenty.
- b) Half of a number minus five gives three.
- c) A number increased by seven is equal to its triple.
- d) The sum of a number and the next consecutive number is fifteen.
Parentheses and Variables on Both Sides
Real equations often have parentheses and the variable on both sides. This order always works:
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Distribute to remove parentheses.
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Combine like terms on each side.
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Get the variable terms on one side (add or subtract a variable term on both sides).
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Finish like a two-step equation.
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:
$4(2x - 3) - 2(5 - 3x) = 3(x + 1) + 5$
Equations with Fractions
Fractions are easier to handle if you get rid of them first. Multiply every term on both sides by the least common denominator (LCD), which is the least common multiple (LCM) of all the denominators. The full routine is below.
Find the value of the unknown $x$ in the following equation with denominators, simplifying beforehand by calculating the lowest common multiple:…
Find the value of the unknown $x$ in the following equation with denominators, simplifying beforehand by calculating 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 the fractions and the use of the lowest common multiple to simplify the…
Solve the following linear equation, paying special attention to the negative signs in front of the fractions and the use of the lowest common multiple to simplify the denominators:
$$\frac{2 \cdot (x - 3)}{5} - \frac{x - 1}{2} = \frac{1}{4} - \frac{3x - 14}{10}$$One Solution, No Solution, or Infinitely Many
Most equations you solve end with $x = \text{a number}$. Two special cases can happen when the variable disappears:
| You end up with | Example | Meaning |
|---|---|---|
| $x = \text{number}$ | $3x = 9$ | One solution: $x = 3$ |
| a false statement | $5 = 9$ | No solution |
| a true statement | $6 = 6$ | Infinitely many solutions (an identity) |
If you graph each side of the equation as a line, the solution is where the lines meet. They can cross once, never meet (parallel), or be the same line.
Compare this with an ordinary equation such as $x + 4 = 10$. It is true only when $x = 6$, so it is not an identity. An equation is an identity only if it is true for every value.
Classify the following mathematical equalities into 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 into identities or equations, briefly explaining why in each case:
- a) $3x + 5 = 20$
- b) $2(x + 1) = 2x + 2$
- c) $x + x = 2x$
- d) $x - 7 = 10$
Word Problems: Four Steps
Most word problems follow the same plan:
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Read and decide what you are looking for.
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Define the variable (say what $x$ means, with units).
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Write an equation from the story.
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Solve, check in the story, and answer with a sentence.
Age problems are easier with a table. Say Priya is $x$ years old, her dad is 4 times as old, and in 6 years the sum of their ages will be 62.
| Now | In 6 years | |
|---|---|---|
| Priya | $x$ | $x + 6$ |
| Dad | $4x$ | $4x + 6$ |
The perimeter of a rectangular padel court is $56$ feet. We know that the length of the court is $2$ feet more than twice its width. Calculate the dimensions of the…
The perimeter of a rectangular padel court is $56$ feet. We know that the length of the court is $2$ feet more than twice its width. Calculate the dimensions of the court by setting up a linear equation.
Marcus has bought three strategy games and two battle passes in a digital shop. Each battle pass costs half as much as a game. If he has paid $\$70…
Marcus has bought three strategy games and two battle passes in a digital shop. Each battle pass costs half as much as a game. If he has paid $\$70$ in total, what is the price of a game and what is the price of a battle pass?
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 currently?
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 currently?
Fractions of a Total, Consecutive Integers, and Two Ways to Count
Three more types of problems come up often.
Fractions of a total. Name the starting amount $x$, then track what is used and what is left. Be careful: "half of the rest" is half of what is left, not half of $x$.
Consecutive integers. If the first is $x$, the next ones are $x + 1$, $x + 2$, and so on.
Two ways to count the same thing. When the same price is reached in two different ways, set the two expressions equal.
The battery level of a mobile phone decreases at a constant rate. During the morning, a quarter of the total charge has been consumed. In the afternoon, a third of what…
The battery level of a mobile phone decreases at a constant rate. During the morning, a quarter of the total charge has been consumed. In the afternoon, a third of what was left after the morning has been used. If there is still $40$% battery remaining ($40$ units of charge), what is the total charge capacity of the phone?
Find three consecutive integers such that the sum of the first and the second exceeds the third number by $15$ units.
Find three consecutive integers such that the sum of the first and the second exceeds the third number by $15$ units.
A group of friends wants to buy a birthday gift that costs a fixed price. If each of them contributes $12 \text{ \$}$, they have $4 \text{ \$}$ left over for wrapping.…
A group of friends wants to buy a birthday gift that costs a fixed price. If each of them contributes $12 \text{ \$}$, they have $4 \text{ \$}$ left over for wrapping. But if each contributes $10 \text{ \$}$, they are $12 \text{ \$}$ short of being able to buy it. How many friends are in the group and how much does the gift cost?
Rates: Distance, Mixtures, and Work
Hard problems usually hide a simple idea. Spot the idea, then write the equation.
Distance, rate, and time
Use $d = rt$. If two travelers move toward each other, their distances add up to the total distance. If one starts later, that traveler has been moving for less time.
Mixtures
When you add water to a solution, the amount of the pure substance does not change. Only the total changes.
Work rates
If a pipe fills a pool in $6$ hours, it fills $\dfrac{1}{6}$ of the pool each hour. When several pipes work together, add their rates. A drain works against you, so its rate is subtracted. In $t$ hours the fractions of the job done must add up to $1$ whole job.
Marcus and Lucy live in different towns located at a distance of $24 \text{ mi}$. At 10:00 a.m., Lucy leaves for Marcus's town on her bicycle at a constant speed of…
Marcus and Lucy live in different towns located at a distance of $24 \text{ mi}$. At 10:00 a.m., Lucy leaves for Marcus's town on her bicycle at a constant speed of $18 \text{ mph}$. Half an hour later, Marcus sets off to meet Lucy on his electric scooter, traveling at $25 \text{ mph}$. At what distance from Lucy's town will they cross paths and at what time will the meeting occur?
In a chemistry laboratory, they want to reduce the concentration of a cleaning solution. We have $12 \text{ liters}$ of a liquid containing $20\%$ pure alcohol. What…
In a chemistry laboratory, they want to reduce the concentration of a cleaning solution. We have $12 \text{ liters}$ of a liquid containing $20\%$ pure alcohol. What quantity 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 filling faucets and a drain. The first faucet can fill the tank in $6 \text{ hours}$, and the second faucet in $4 \text{ hours}$. If the tank is…
A water tank has two filling faucets and a drain. The first faucet can fill the tank in $6 \text{ hours}$, and the second faucet in $4 \text{ hours}$. If the tank is full, the drain can empty it completely in $8 \text{ hours}$. If we open both filling faucets and the drain at the same time with the tank empty, how long will it take to fill completely?
Challenge Problems: Changing Shapes, Ages, and Digits
In these problems a quantity changes, and you must write the new quantity in terms of $x$ before you form the equation.
For digit problems, call the digit you know the most about $x$ and write the others in terms of it. If the tens digit is 3 times the ones digit and the digits add to 8, then with ones digit $x$: $x + 3x = 8$, so $x = 2$ and the number is $62$. Always finish by writing the number, not just $x$.
A graphic design company creates rectangular posters. In their latest model, the base measures $10 \text{ cm}$ more than the height. To comply with a new standard, they…
A graphic design company creates rectangular posters. In their latest model, the base measures $10 \text{ cm}$ more 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 perimeter of the new poster is $110 \text{ cm}$, what were the original dimensions of the poster?
A father's current age is three times that of his son. Five years ago, the sum of their ages was the same as the age the father will be in ten years. 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 the same as the age the father will be in ten years. How old are the father and son currently?
Find a three-digit number with the following characteristics: the tens digit is twice the ones digit, and the hundreds digit is three times the ones digit. If we add the…
Find a three-digit number with the following characteristics: the tens digit is twice the ones digit, and the hundreds digit is three times the ones digit. If we add the three digits, the result is the same as if we subtract the ones digit from the hundreds digit and add 10. Once the equation is set up, state what the number is. (Hint: call the ones digit $x$).
Back to the start: With $x$ as the price of one taco, $3x + 1 = 10$ gives $3x = 9$, so $x = 3$. Each taco costs $\$3$.