Personal Finance – Common Scenarios: Background You’ll Need – Page 2

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Personal Finance – Common Scenarios: Background You’ll Need – Page 2

Solve complex equations

Solving Complex Equations

You are likely to encounter equations that are more challenging than the basic linear ones. These complex equations can involve multiple variables, higher powers, or intricate combinations of terms. The key to mastering these equations lies in recognizing patterns and understanding how to manipulate the terms to reveal the solutions.

When solving complex equations:

Identify the structure of the equation and plan your approach.
Look for ways to simplify the equation, such as combining like terms or factoring.
Be methodical in your application of algebraic rules to avoid common pitfalls.

Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. This will help us with organization. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to isolate the variable terms on one side of the equation.

Solve the following for [latex]x[/latex]:

[latex]7x+5=6x+2[/latex]

Solve the following for [latex]n[/latex]:

[latex]6n - 2=-3n+7[/latex]

	[latex]7x+5=6x+2[/latex]
Collect the variable terms to the left side by subtracting [latex]6x[/latex] from both sides.	[latex]7x(\color{red}{-6x}\color{black})+5=6x(\color{red}{-6x}\color{black})+2[/latex]
Simplify.	[latex]x+5=2[/latex]
Now, collect the constants to the right side by subtracting [latex]5[/latex] from both sides.	[latex]x+5(\color{red}{-5}\color{black})=2(\color{red}{-5}\color{black})[/latex]
Simplify.	[latex]x=-3[/latex]
The solution is [latex]x=-3[/latex] .
Check:	[latex]7x+5=6x+2[/latex] Let [latex]x=-3[/latex] [latex]7(\color{red}{-3}\color{black})+5\stackrel{\text{?}}{=}6(\color{red}{-3}\color{black})+2[/latex] [latex]-21+5\stackrel{\text{?}}{=}-18+2[/latex] [latex]16=16\quad\checkmark[/latex]

	[latex]6n-2=-3n+7[/latex]
We don’t want variables on the right side—add [latex]3n[/latex] to both sides to leave only constants on the right.	[latex]6n(\color{red}{+3n}\color{black})-2=-3n(\color{red}{+3n}\color{black})+7[/latex]
Combine like terms.	[latex]9n-2=7[/latex]
We don’t want any constants on the left side, so add [latex]2[/latex] to both sides.	[latex]9n-2(\color{red}{+2}\color{black})=7(\color{red}{+2}\color{black})[/latex]
Simplify.	[latex]9n=9[/latex]
The variable term is on the left and the constant term is on the right. To get the coefficient of [latex]n[/latex] to be one, divide both sides by [latex]9[/latex].	[latex]\Large\frac{9n}{\color{red}{9}\color{black}}\normalsize =\Large\frac{9}{\color{red}{9}\color{black}}[/latex]
Simplify.	[latex]n=1[/latex]
Check:	[latex]6n-2=-3n+7[/latex] Substitute [latex]1[/latex] for [latex]n[/latex] [latex]6(\color{red}{1}\color{black})-2\stackrel{\text{?}}{=}-3(\color{red}{1}\color{black})+7[/latex] [latex]4=4\quad\checkmark[/latex]