An equation is a mathematical statement with two expressions that have the same value. The two expressions are separated by an equal sign.
Whatever we do to one side of the equation, we must do the exact same to the other side of the equation to keep the balanced and equal.
Correctly solving an equation gives us the solution, also referred to as the root.
A one-step equation is an algebraic equation you can solve in just one step. You've solved the equation when you isolate the variable (get the variable by itself on one side of the equation).
This can be achieved by performing the opposite operation occuring on the variable. Remember, operations are simply \(+, -, *,\) or \(/\). Addition/Subtration and Multiplication/Division are opposite operations. You must apply the operation to both sides of the equation!
We can solve the following equation for the variable \(x\). That means, we can solve for the value of \(x\) that makes this equation true.
\(\cfrac{x}{3} = -8\)
Since \(x\) is being divided by \(3\) on the left side, we need to do the opposite (multiply by \(3\)) on both sides:
\(\cfrac{x}{3} \cdot \textcolor{red}{3} = -8 \cdot \textcolor{red}{3}\)
On the left side, we have \(x\) divided by \(3\) and multiplied by \(3\). These operations cancel leaving just \(x\). It is isolated and we can solve for the answer:
\(x = -24\)
Our answer means that this equation is true when \(x = -24\). You can check your answer by substituting \(x = -24\) into the original equation:
\(\cfrac{x}{3} = -8\)
\(\cfrac{\textcolor{red}{-24}}{3} = -8\)
\(-8 = -8\)
Therefore, we can confirm that \(\boldsymbol{x = -24}\).
A two-step equation is similar to a one step equation, but invovles one extra step. Most of the time they would take in one of the three forms:
| \(a \cdot x + b = c\) | \(\cfrac{x}{a} + b = c\) | \(a \cdot ( x + b ) = c\) |
* Distributive Property states that \(a(x+b)\) equals \(a \cdot x + a \cdot b\)
When you have a two-step equation, try the following steps:
NOTE: These operations are summarized under the acronym BEDMAS.
First, we can subtract \(3\) from both sides of the equation:
\(4x + 3 \textcolor{red}{-3} = 19 \textcolor{red}{-3}\)
\(4x = 16\)
Next, we can divide both sides of the equation by \(4\):
\(\cfrac{4 x}{\textcolor{red}{4}} = \cfrac{16}{\textcolor{red}{4}}\)
\(x = 4\)
Then, we can plug \(x = 4\) into the original equation to confirm our results:
\(4x + 3 = 19\)
\(4(\textcolor{red}{4}) + 3 = 19\)
\(16 + 3 = 19\)
\(19 = 19\)
Therefore, we can confirm that \(\boldsymbol{x = 19}\).
First, we can apply distributive property:
\(4(\textcolor{red}{3x}) = 600\)
\(12x = 600\)
Next, we can divide both sides of the equation by \(12\):
\(\cfrac{12x}{\textcolor{red}{12}} = \cfrac{600}{\textcolor{red}{12}}\)
\(x = 50\)
Then, we can plug \(x = 50\) into the original equation to confirm our results:
\(4(3x) = 600\)
\(4(3(\textcolor{red}{50})) = 600\)
\(4(150) = 600\)
\(600 = 600\)
Therefore, we can confirm that \(\boldsymbol{x = 50}\).
Similiar to two step equations, but again with the addition of an extra step to solve.
First, we can add \(1\) to both sides of the equation:
\(\cfrac{2x}{5} - 1 \textcolor{red}{+1} = 7 \textcolor{red}{+1}\)
\(\cfrac{2x}{5} = 8\)
Next, we can multiply both sides by \(5\):
\(\textcolor{red}{5} \cdot \cfrac{2x}{5} = \textcolor{red}{5} \cdot 8\)
\(2x = 40\)
Then, we can divide both sides by \(2\):
\(\cfrac{2 \cdot x}{\textcolor{red}{2}} = \cfrac{40}{\textcolor{red}{2}}\)
\(x = 20\)
After, we can plug \(x = 20\) into the original equation to confirm our results:
\(\cfrac{2x}{5} - 1 = 7\)
\(\cfrac{2(\textcolor{red}{20})}{5} - 1 = 7\)
\(\cfrac{40}{5} - 1 = 7\)
\(8 - 1 = 7\)
\(7 = 7\)
Therefore, we can confirm that \(\boldsymbol{x = 20}\).
Sometimes you may come across \(x\) in the denominator of a fraction or even fractions on both side of the equation. In these cases, one way to start solving the question is to cross-multiply.
The general rule for cross multiply is:
if: \(\cfrac{a}{b} = \cfrac{c}{d}\), then: \(a \cdot d = b \cdot c\)
Don't forget that when you have a whole number, you can make it a fraction by putting a \(1\) in the denominator.
First we can cross-multiply the \(2\) expressions :
\(3 \cdot x = 2 \cdot 8\)
\(3x = 16\)
Next, we can divide both sides by \(3\):
\(\cfrac{3x}{\textcolor{red}{3}} = \cfrac{16}{\textcolor{red}{3}}\)
\(x = \cfrac{16}{3}\)
Then, we can plug \(x = \cfrac{16}{3}\) into the original equation to confirm our results:
\(\cfrac{3}{8} = \cfrac{2}{x}\)
\(\cfrac{3}{8} = \cfrac{2}{\textcolor{red}{\cfrac{16}{3}}}\)
\(\cfrac{3}{8} = 2 \left(\cfrac{3}{16}\right)\)
\(\cfrac{3}{8} = \cfrac{6}{16}\)
\(\cfrac{3}{8} = \cfrac{3}{8}\)
Therefore, we can confirm that \(\boldsymbol{x = \cfrac{16}{3}}\).