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Hello! This module rebuilds the everyday algebra you will need before working with vectors, matrices, and the mathematics behind language models. Today’s goal is specific: take an equation with one unknown and get that unknown by itself. We will do it by hand first, so each step has a reason rather than feeling like a rule to memorize.
An equation is a promise of equal value
Consider . The left and right sides do not look the same, but the equals sign says they have the same value. Solving means finding a value of that makes that promise true. Isolating means rewriting the equation until it says a number.
The key rule is simple: if you change one side of an equation, make the same change to the other side. Otherwise, you may break the equality. Watch how the balance-scale picture and a first example make this rule concrete.
Algebra Basics: Solving Basic Equations Part 1 - Math Antics
Watch Algebra Basics: Solving Basic Equations Part 1 from Math Antics for the meaning of “isolate” and the reason each operation must be done on both sides.
Watch the balance idea first. Then watch adding example, where subtracting 7 frees the unknown, and subtraction example, where adding 5 does the reverse. Focus on what stays equal at every step.
For instance, has 7 added to . Subtract 7 from both sides:
Since , this becomes . The value is easy to check in the original equation: .
This is the useful meaning of an inverse operation: subtraction undoes addition, and addition undoes subtraction. You are not magically “moving” a number across the equals sign; you are performing a legal operation on both sides.
Multiplication and division work the same way
In , the notation means multiplied by . To leave just , undo multiplication by 3 using division by 3 on both sides:
So . Likewise, if , undo division by 4 by multiplying both sides by 4; then .
Algebra Basics: Solving Basic Equations Part 2 - Math Antics
Watch Algebra Basics: Solving Basic Equations Part 2 from Math Antics for the multiplication-and-division counterparts to the previous video.
Begin with inverse operations. Watch dividing both sides to see why 3x=15 gives x=5. Then use multiplying both sides to see how to handle an unknown divided by a number. The important distinction is what is happening to the unknown, not which arithmetic symbol you notice first.
There is one small safety rule: never divide both sides by zero. Division by zero is undefined. In the examples above, the numbers we divide by are nonzero.
Undo the outer operation first
Many equations need two steps. Look at . The unknown is multiplied by 6, then 2 is added. To free , undo those actions in reverse order: remove the added 2 first, then remove the multiplication by 6.
Here is the left-hand example written out, including the operation on both sides:
That simplifies to . Now divide both sides by 6:
Thus . Check: .
The right-hand equation in the image works the same way, although its first inverse operation is different. For , add 5 to both sides to get . Dividing both sides by 4 gives . The letter does not change the method.
You can use the same reasoning when the numbers are less tidy. In , subtracting 5 from both sides gives . Dividing both sides by gives . The check, , confirms that the negative answer is not a mistake.
Notice the pattern rather than memorizing these answers: identify what was done to the unknown, undo it on both sides, and simplify.
What if the unknown appears on both sides?
Sometimes isolating the unknown first means gathering its copies together. In , subtract from both sides:
The right-hand terms cancel, leaving . Divide both sides by 3 to get . A check in the original equation gives ; both sides equal 15.
Parentheses can also hide a straightforward equation. For example, can be solved by dividing both sides by 2 first, giving , then subtracting 3 to get . Alternatively, you could expand the parentheses first. Either route is valid if you keep the equality intact.
The short reading below brings the balance rule, a two-step equation, an unknown on both sides, and parentheses together in one place.
Read Solving Equations from Math is Fun to see how the same balance rule handles several equation layouts.
Under “How to Solve an Equation,” start at the sentence about the Golden Rule. Read the two-step explanation through its conclusion that x=5. Then, under “Variables on Both Sides,” read the next two cases: follow the 5x=2x+9 example through x=3, and continue under “Using Parentheses” to the worked 3(x+2)=15 example, ending at x=3. Focus on the operation applied to both sides at each line, rather than on the phrase “move terms.”
One final detail helps you recognize unusual equations. If subtracting the unknown terms makes them all disappear, do not try to divide by their now-zero coefficient. For example, would imply , so no value of works. By contrast, is true for every value of . Most equations in this lesson have one answer, but checking what remains after cancellation tells you when they do not.
A reliable way to finish
For a one-variable linear equation, keep this compact routine in mind:
- Decide what prevents the unknown from standing alone. If it appears on both sides, gather its terms onto one side first.
- Undo operations by applying the same operation to both sides. With two steps, usually undo addition or subtraction before multiplication or division.
- Substitute your answer into the original equation. The two sides should have equal values.
You can now read not as a lucky guess but as the result of equality-preserving steps. In the next lesson, you will use this skill in the other direction: turn a short real-world description into an equation, then solve it.