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How to Learn Algebra (with Pictures) - wikiHow
How to Learn Algebra
Co-authored by Daron Cam
Last Updated: January 27, 2023
Approved
Learning algebra can seem intimidating, but once you get the hang of it, it’s not that hard!
You just have to follow the order for completing parts of the equation and keep your work
organized to avoid mistakes!
Part
1
Part 1 of 5:
Learning Basic Algebra Rules
1
Review your basic math operations. To start learning algebra, you'll need to
know basic math skills such as adding, subtracting, multiplying and
dividing.[1] This primary/elementary school math is essential before you start
learning algebra.[2] If you don't have these skills mastered, it will be tricky to
tackle the more complex concepts taught in algebra. If you need a refresher on
these operations, try our article on basic math skills.
You don't necessarily need to be great at doing these basic operations in your
head to do algebra problems. Many algebra classes will allow you to use a
calculator to save time when doing these simple operations. You should,
however, at least know how to do these operations without a calculator for
when you aren't allowed to use one.
Know the order of operations. One of the trickiest things about solving an
algebra equation as a beginner is knowing where to start. Luckily, there's a
specific order for solving these problems: first do any math operations in
parentheses, then do exponents, then multiply, then divide, then add, and finally
subtract. A handy tool for remembering this order of operations is the acronym
2
PEMDAS.[3] Learn how to apply the order of operations here. To recap, the order
of operations is:
Parentheses
Exponents
Multiplication
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Division
Addition
Subtraction
The order of operations is important in algebra because doing the operations
in an algebra problem in the wrong order can sometimes affect the answer.
For instance, if we're dealing with the math problem 8 + 2 × 5, if we add 2 to 8
first, we get 10 × 5 = 50, but if we multiply 2 and 5 first, we get 8 + 10 = 18.
Only the second answer is correct.
3
Know how to use negative numbers. In algebra, it's common to use
negative numbers, so it's smart to review how to add, subtract, multiply, and
divide negatives before starting to learn algebra.[4] Below are just a few negative
number basics to keep in mind — for more information, see our articles on adding
and subtracting negative numbers and dividing and multiplying negative numbers.
On a number line, a negative version of a number is the same distance from
zero as the positive, but in the opposite direction.
Adding two negative numbers together makes the number more negative (in
other words, the digits will be higher, but since the number is negative, it
counts as being lower)
Two negative signs cancel out — subtracting a negative number is the same
as adding a positive number
Multiplying or dividing two negative numbers gives a positive answer.
Multiplying or dividing a positive number and a negative number gives a
negative answer.
Know how to keep long problems organized. While simple algebra
problems can be a snap to solve, more complicated problems can take many,
many steps. To avoid errors, keep your work organized by starting a new line
every time you make a step toward solving your problem. If you're dealing with a
two-sided equation, try to write all the equals signs ("="s) underneath each
4
other.[5] This way, if you make a mistake somewhere, it'll be much easier to find
and correct.
For example, to solve the equation 9/3 - 5 + 3 × 4, we might keep our problem
organized like this:
9/3 - 5 + 3 × 4
9/3 - 5 + 12
3 - 5 + 12
3+7
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Part
2
How to Learn Algebra (with Pictures) - wikiHow
Part 2 of 5:
Understanding Variables
Look for symbols that aren't numbers. In algebra, you'll start to see letters
and symbols appear in your math problems, rather than just numbers. These
are called variables. Variables aren't as confusing as they may first seem - they're
1
just ways of showing numbers with unknown values.[6] Below are just a few
common examples of variables in algebra:
Letters like x, y, z, a, b, and c
Greek letters like theta, or θ
Note that not all symbols are unknown variables. For instance, pi, or π, is
always equal to about 3.14159.
2
Think of variables as "unknown" numbers. As mentioned above, variables
are basically just numbers with unknown values.[7] In other words, there's
some number that can go in place of the variable to make the equation work.
Usually, your goal in an algebra problem is to figure out what the variable is —
think of it as a "mystery number" that you're trying to discover.
For example, in the equation 2x + 3 = 11, x is our variable. This means that
there's some value that goes in the place of x to make the left side of the
equation equal 11. Since 2 × 4 + 3 = 11, in this case, x = 4.
An easy way to start understanding variables is to replace them with question
marks in algebra problems. For example, we might re-write the equation 2 + 3
+ x = 9 as 2 + 3 + ? = 9. This makes it easier to understand what we're trying
to do — we just need to find out what number to add to 2 + 3 = 5 to get 9. The
answer is again 4, of course.
Watch for recurring variables. If a variable appears more than once, simplify
the variables. What do you do if the same variable appears more than once in
the equation? Though this situation may seem tricky to solve, you can actually
treat variables how you'd treat normal numbers — in other words, you can add
them, subtract them, and so on as long as you only combine variables that are
alike. In other words, x + x = 2x, but x + y doesn't equal 2xy.
3
For example, let's look at the equation 2x + 1x = 9. In this case, we can add
2x and 1x together to get 3x = 9. Since 3 x 3 = 9, we know that x = 3.
Note again that you can only add the same variables together. In the equation
2x + 1y = 9, we can't combine 2x and 1y because they are two different
variables.
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This is also true for when one variable has a different exponent than another.
For instance, in the equation 2x + 3x2 = 10, we can't combine 2x and 3x2
because the x variables have different exponents. See How to Add Exponents
for more information.
Part
3
Part 3 of 5:
Learning to Solve Equations by "Canceling"
Try to get the variable by itself in algebra equations. Solving an equation
in algebra usually means finding out what the variable is. Algebra equations
are usually set up with numbers and/or variables on both sides, like this: x + 2 = 9
× 4. To figure out what the variable is, you need to get it by itself on one side of
the equals sign. Whatever is left on the other side of the equals sign is your
answer.
1
In the example (x + 2 = 9 × 4), to get x by itself on the left side of the equation,
we need to get rid of the "+ 2". To do this, we'll simply subtract 2 from that
side, leaving us with x = 9 × 4. However, to keep both sides of the equation
equal, we also need to subtract 2 from the other side. This leaves us with x =
9 × 4 - 2. Following the order of operations, we multiply first, then subtract,
giving us an answer of x = 36 - 2 = 34.
Cancel addition with subtraction (and vice versa). As we just saw above,
getting x by itself on one side of the equals sign usually means getting rid of
the numbers next to it. To do this, we perform the "opposite" operation on both
sides of the equation. For instance, in the equation x + 3 = 0, since we see a "+ 3"
next to our x, we'll put a "- 3" on both sides. The "+ 3" and "- 3", leaving x by itself
and "-3" on the other side of the equals sign, like this: x = -3.
2
In general, addition and subtraction are like "opposites" — do one to get rid of
the other. See below:
For addition, subtract. Example: x + 9 = 3 → x = 3 - 9
For subtraction, add. Example: x - 4 = 20 → x = 20 + 4
Cancel multiplication with division (and vice versa). Multiplication and
division are a little harder to work with than addition and subtraction, but they
have the same "opposite" relationship. If you see a "× 3" on one side, you'll cancel
it by dividing both sides by 3, and so on.
3
With multiplication and division, you must perform the opposite operation on
everything on the other side of the equals sign, even if it's more than one
number. See below:
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For multiplication, divide. Example: 6x = 14 + 2→ x = (14 + 2)/6
For division, multiply. Example: x/5 = 25 → x = 25 × 5
Cancel exponents by taking the root (and vice versa). Exponents are a
fairly advanced pre-algebra topic — if you don't know how to do them, see our
basic exponent article for more information. The "opposite" of an exponent is the
4
root that has the same number as it. For example, the opposite of the 2 exponent
is a square root (√), the opposite of the 3 exponent is the cube root (3√), and so
on.[8]
It may be a little confusing, but, in these cases, you take the root of both sides
when dealing with an exponent. On the other hand, you take the exponent of
both sides when you're dealing with a root. See below:
For exponents, take the root. Example: x2 = 49 → x = √49
For roots, take the exponent. Example: √x = 12 → x = 122
Part
4
Part 4 of 5:
Sharpening Your Algebra Skills
Use pictures to make problems clearer. If you're having a hard time
visualizing an algebra problem, try using diagrams or pictures to illustrate your
equation. You can even try using a group of physical objects (like blocks or coins)
1
instead if you have some handy.[9]
For example, let's solve the equation x + 2 = 3 by using boxes (☐)
x +2 = 3
☒+☐☐ =☐☐☐
At this point, we'll subtract 2 from both sides by simply removing 2 boxes
(☐☐) from both sides:
☒+☐☐-☐☐ =☐☐☐-☐☐
☒=☐, or x = 1
As another example, let's try 2x = 4
☒☒ =☐☐☐☐
At this point, we'll divide both sides by two by separate the boxes on each
side into two groups:
☒|☒ =☐☐|☐☐
☒ = ☐☐, or x = 2
2
Use "common sense checks" (especially for word problems). When
converting a word problem into algebra, try to check your formula by plugging
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in simple values for your variable. Does your equation make sense when x=0?
When x=1? When x = -1? It's easy to make simple mistakes by writing down p=6d
when you mean p=d/6, but these are easily caught if you do a quick sanity check
on your work before going further.
For example, let's say we're told that a football field is 30 yards (27.4 m)
longer than it is wide. We use the equation l = w + 30 to represent this. We
can test whether this equation makes sense by plugging in simple values for
w. For instance, if the field is w = 10 yards (9.1 m) wide, it will be 10 + 30 = 40
yards (36.6 m) long. If it's 30 yards (27.4 m) wide, it will be 30 + 30 = 60 yards
(54.9 m) long, and so on. This makes sense — we'd expect the field to get
longer as it gets wider, so this equation is reasonable.
Be aware that answers won't always be integers in algebra. Answers in
algebra and other advanced forms of math aren't always round, easy
numbers. They can often be decimals, fractions, or irrational numbers. A calculator
can help you find these complicated answers, but keep in mind that your teacher
may require you to give your answer in its exact form, not in an unwieldy decimal.
3
For instance, let's say that we narrow down an algebra equation to x = 12507.
If we type 12507 into a calculator, we'll get a huge string of decimals (plus,
since the calculator's screen is only so large, it can't display the entire
answer.) In this case, we may want to represent our answer as simply 12507
or else simplify the answer by writing it in scientific notation.
Try expanding your skill. When you're confident with basic algebra, try
factoring. One of the trickiest algebra skills of all is factoring — a sort of
shortcut for getting complex equations into simple forms. Factoring is a semiadvanced algebra topic, so consider consulting the article linked above if you're
having trouble mastering it. Below are just a few quick tips for factoring equations:
4
Equations with the form ax + ba factor to a(x + b). Example: 2x + 4 = 2(x + 2)
Equations with the form ax2 + bx factor to cx((a/c)x + (b/c)) where c is the
biggest number that divides into a and b evenly. Example: 3y2 + 12y = 3y(y +
4)
Equations with the form x2 + bx + c factor to (x + y)(x + z) where y × z = c and
yx + zx = bx. Example: x2 + 4x + 3 = (x + 3)(x + 1).
Practice, practice, practice! Progress in algebra (and any other kind of
math) requires lots of hard work and repetition. Don't worry — by paying
attention in class, doing all of your assignments, and seeking out help from your
5
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teacher or other students when you need it, algebra will begin to become second
nature.
Ask your teacher to help you understand tricky algebra topics. If you're
having a hard time getting the hang of algebra, don't worry — you don't have
to learn it on your own. Your teacher is the first person you should turn to with
questions. After class, politely ask your teacher for help. Good teachers will
usually be willing to re-explain the day's topic at an after-school appointment and
6
may even be able to give you extra practice materials.[10]
If, for some reason, your teacher can't help you, try asking them about tutoring
options at your school.[11] Many schools will have some sort of after-school
program that can help you get the extra time and attention you need to start
excelling at your algebra. Remember, using free help that's available to you
isn't something to be embarrassed about — it's a sign that you're smart
enough to solve your problem!
Part
5
Part 5 of 5:
Exploring Intermediate Topics
1
Learn to graph x/y equations. Graphs can be valuable tools in algebra
because they allow you to display ideas that you'd usually need numbers for
in easy-to-understand pictures.[12] Usually, in beginning algebra, graphing
problems are restricted to equations with two variables (usually x and y) and are
done on a simple 2-D graph with an x axis and a y axis. With these equations, all
you need to do is plug in a value for x, then solve for y (or do the reverse) to get
two numbers that correspond to a point on the graph.
For example, in the equation y = 3x, if we plug in 2 for x, we get y = 6. This
means that the point (2,6) (two spaces to the right of center and six spaces
above center) is part of this equation's graph.
Equations with the form y = mx + b (where m and b are numbers) are
especially common in basic algebra. These equations always have a slope of
m and cross the y axis at y = b.
Learn to solve inequalities. What do you do when your equation doesn't use
an equals sign? Nothing much different than what you'd normally do, it turns
out. For inequalities, which use signs like > ("greater than") and < ("less than"),
just solve as normal. You'll be left with an answer that's either less than or greater
than your variable.
2
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For instance, with the equation 3 > 5x - 2, we would solve just like we would
for a normal equation:
3 > 5x - 2
5 > 5x
1 > x, or x < 1.
This means that every number less than one works for x. In other words, x can
be 0, -1, -2, and so on. If we plug these numbers into the equation for x, we'll
always get an answer less than 3.
3
Tackle quadratic equations. One algebra topic that many beginners struggle
with is solving quadratic equations. Quadratics are equations with the form
ax2 + bx + c = 0, where a, b, and c are numbers (except that a can't be 0.) These
equations are solved with the formula x = [-b +/- √(b2 - 4ac)]/2a . Be careful — the
+/- sign means you need to find the answers for adding and subtracting, so you
can have two answers for these types of problems.
As an example, let's solve the quadratic formula 3x2 + 2x -1 = 0.
x = [-b +/- √(b2 - 4ac)]/2a
x = [-2 +/- √(22 - 4(3)(-1))]/2(3)
x = [-2 +/- √(4 - (-12))]/6
x = [-2 +/- √(16)]/6
x = [-2 +/- 4]/6
x = -1 and 1/3
Experiment with systems of equations. Solving more than one equation at
once may sound super-tricky, but when you're working with simple algebra
equations, it's not actually that hard. Often, algebra teachers use a graphing
approach for solving these problems. When you're working with a system of two
equations, the solutions are the points on a graph that the lines for both equations
cross at.
4
For example, let's say we're working with a system that contains the equations
y = 3x - 2 and y = -x - 6. If we draw these two lines on a graph, we get one line
that goes up at a steep angle, and one that goes down at a mild angle. Since
these lines cross at the point (-1,-5), this is a solution to the system.[13]
If we want to check our problem, we can do this by plugging our answer into
the equations in the system — a right answer should "work" for both.
y = 3x - 2
-5 = 3(-1) - 2
-5 = -3 - 2
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-5 = -5
y = -x - 6
-5 = -(-1) - 6
-5 = 1 - 6
-5 = -5
Both equations "check out," so our answer is right!
Expert Q&A
Question
What are the basics of algebra?
Daron Cam
Academic Tutor
Expert Answer
Basic math skills you learned in elementary or primary school are the
fundamentals of algebra. This includes concepts like adding, subtracting,
multiplying and dividing.
Question
How do I solve x + 13 = 24?
Community Answer
Subtract 13 from both sides to get x by itself. That makes the equation: x = 24 13 or: x = 11.
Question
Would 8X + 9 be the same as 8x X 9?
Community Answer
No, because the first equation asks for addition and the second equation asks for
multiplication.
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See more answers
Tips
There are tons of resources for people learning algebra online. For instance, just
a simple search engine query like "algebra help" can yield dozens of great results.
You may also want to try browsing WikiHow's selection of math articles. There's a
huge amount of information out there, so start exploring today!
One great site for algebra beginners is khanacademy.com. This free site offers
tons of easy-to-follow lessons on a huge variety of topics, including algebra. There
are videos for everything from the extreme basics to advanced university-level
topics, so don't be afraid to dive in to Khan Academy's material and start using all
the help that the site has to offer!
Don't forget that your best resources when you're trying to learn algebra can be
the people you're already comfortable with. Try talking to friends or fellow students
who are taking the class with you if need extra help understanding your last
lesson.
Show More Tips
References
1.
2.
3.
4.
5.
6.
7.
8.
https://www.basic-mathematics.com/basic-operations.html
Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
https://www.mathsisfun.com/operation-order-pemdas.html
https://www.mathsisfun.com/definitions/negative.html
https://web.cs.ucdavis.edu/~amenta/w10/writingman.pdf
https://www.techopedia.com/definition/1816/variable-mathematics
https://www.britannica.com/topic/variable-mathematics-and-logic
http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_R
ESOURCE/U16_L1_T3_text_final.html
9. https://www.smartickmethod.com/blog/math/operations-and-algebraicthinking/algebra/singapore-bars-algebraic-equations/
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10. Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
11. Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
12. https://www.khanacademy.org/math/algebra-basics/alg-basics-graphing-lines-andslope
13. http://www.purplemath.com/modules/systlin1.htm
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