# Simple Equations

When solving a **simple equation**, it is helpful to think of the equation as a balance, with the equals sign (\(=\)) being the fulcrum or center. Thus, if an operation is performed one side of the equation, the same must be done on the other side. Doing the same thing to both sides of the equation keeps the equation balanced. Just as adding masses of \(10\text{ kg}\) to both sides of a beam keeps it balanced, so too does adding \(10\) to both sides of an equation.

Solving an equation is the process of getting what you're looking for, or solving for, on one side of the equals sign and everything else on the other side. You're really sorting information. If you're solving for \(x\), you must get \(x\) on one side by itself.

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## Addition and Subtraction Equations

Some equations involve only addition and/or subtraction.

## Solve for \(x\)

\[x+8=12.\]

To solve the equation \(x + 8 = 12\), you must get \(x\) by itself on one side. Therefore, subtract \(8\) from both sides to obtain

\[\begin{align} x+8-8&=12-8\\ x&=4. \ _\square \end{align}\]

## Solve for \(y\)

\[y-19=28.\]

To solve this equation, you must get \(y\) by itself on one side. Therefore, just move the \(-19\) to the other side and change the sign to obtain

\[\begin{align} y&=28+19\\ &=47. \ _\square \end{align}\]

Solve the following system of equations: \[\begin{align} 3x - 2y &= 0\\ 17x - 7y &= 13. \end{align}\]

There are two ways to solve this problem.

Solution 1:We are given \[\begin{align} 3x - 2y &= 0 &\qquad (1)\\ 17x - 7y &= 13, &\qquad (2)\\ \end{align}\] and we need to find the values of \(x\) and \(y.\) Now, from equation (1), we get \[3x - 2y = 0 \implies 3x = 2y \implies x = \frac{2y}{3}. \qquad (3)\] Substituting (3) into (1) gives \[\begin{align} 17 \times \frac{2y}{3} - 7y &= 13\\ \frac{34y}{3} - 7y &= 13\\ \frac{34y - 21y}{3} &= 13\\ \frac{13y}{3}&= 13\\ 13y &= 39\\ y &= \frac{39}{13} = 3. \end{align}\] Substituting this into (3) gives \[x = \frac{2 × 3}{3}=2.\] Therefore, \((x,y)= ( 2,3).\) \(_\square\)

Solution 2:We are given \[\begin{align} 3x - 2y &= 0 &\qquad (1)\\ 17x - 7y &= 13, &\qquad (2)\\ \end{align}\] and we need to find the values of \(x\) and \(y.\) Now, multiplying (1) by 7 and (2) by 2 and then subtracting gives \[\begin{align} (1)\times 7 = 21x - 14y &= 0 \\ (2)\times 2 = 34x - 14y &= 26\ \ (- \\ \hline 13x &= 26\\ x &= 2. \end{align}\] Substituting this into (1) gives \[3\times 2-2y = 0 \implies y=3.\] Therefore, \((x,y)= ( 2,3).\) \(_\square\)

## Multiplication and Division Equations

Some equations involve only multiplication or division. This is typically when the variable is already on one side of the equation, but there is either more than one of the variable, such as \(2x\), or a fraction of the variable, such as \(\dfrac{1}{3} x\) or \(\dfrac {x}{2}.\)

In the same manner as when you add or subtract, you can multiply or divide both sides of an equation by the same number, as long as it is not **zero**, and the equation will not change.

## Solve for \(x\)

\[3x = 9.\]

To deal with the coefficient of 3, divide each side of the equation by \(3\) to obtain

\[\begin{align} \dfrac {3x}{3} &= \dfrac {9}{3}\\ x &= 3. \ _\square \end{align}\]

## Solve for \(y\)

\[\dfrac{y}{7} = 28.\]

To deal with the division by 7, multiply each side by \(7\) to obtain

\[\begin{align} \dfrac {7y}{7} &= 196\\ y&=196. \ _\square \end{align}\]

## Solve for \(x\)

\[\dfrac{3}{4} x = 18.\]

To deal with the coefficient of \( \frac{3}{4} \), multiply each side by the reciprocal \(\dfrac {4}{3}\), to obtain

\[\begin{align} \left(\dfrac {4}{3} \right) \left(\dfrac {3}{4} x \right) &= \left(\dfrac {4}{3} \right) 18\\ x &= 24. \ _\square \end{align}\]

## Multi-step Equations

Main Article: Multi-step equations

Sometimes equations require more than one step in order to solve for the desired value. It is helpful to think of the variable that we wish to isolate as trapped behind multiple chests, each with a different lock. Here, each lock represents an operation that can be **unlocked** with the appropriate inverse operation (+ unlocks -, while \(\times\) unlocks \(\div\), and \(^2\) unlocks \(\sqrt{\text{ }}\)). With each operation applied to both sides of an equation, a lock is removed, thereby moving us closer to the desired variable.

## What value of \(x\) satisfies \(\sqrt{3x+7}=4?\)

Solving for \(x\), we have: \[\begin{align} \sqrt{3x + 7} &= 4 &\qquad (\text{Write the equation})\\

3x + 7 &= 16 &\qquad (\text{Square both sides})\\

3x &= 9 &\qquad (\text{Subtract 7 from both sides})\\

x &= 3.\ _\square &\qquad (\text{Divide both sides by 3}) \end{align}\]

**Cite as:**Simple Equations.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/solving-equations-simple/