Personal Finance – Common Scenarios: Background You’ll Need – Page 1

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Personal Finance – Common Scenarios: Background You’ll Need – Page 1

Solve linear equations

Solving Linear Equations

Before we delve into solving linear equations, it’s essential to grasp the basic principles behind them. A linear equation is an algebraic statement where two expressions are set equal to each other, often involving a constant and a variable. The goal is to find the value of the variable that makes the equation true. These equations are fundamental in algebra and provide a gateway to understanding more complex mathematical concepts.

How to: Solve a linear equation:

Begin by simplifying the expressions on each side of the equation as much as possible.
Use inverse operations to isolate the variable on one side of the equation.
Perform the same operations on both sides to maintain equality.

Remember, solving linear equations is not just about finding an answer—it’s about understanding the relationship between variables and constants, and how changes in one affect the other.

Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation first. You should always simplify as much as possible before trying to isolate the variable.

Solve:

[latex]3x - 7 - 2x - 4=1[/latex]

Solve:

[latex]2\left(3k - 1\right)-5k=-2 - 7[/latex]

	[latex]3x-7-2x-4=1[/latex]
Rearrange the terms, using the Commutative Property of Addition.	[latex]3x-2x-7-4=1[/latex]
Combine like terms.	[latex]x-11=1[/latex]
Add [latex]11[/latex] to both sides to isolate [latex]x[/latex] .	[latex]x-11(\color{red}{+11}\color{black})=1(\color{red}{+11}\color{black})[/latex]
Simplify.	[latex]x=12[/latex]
Check.	Substitute [latex]x=12[/latex] into the original equation. [latex]\begin{array}{rcl} 3x-7-2x-4 & = & 1 \\ 3(\color{red}{12}\color{black})-7-2(\color{red}{12}\color{black})-4 & = & 1 \\ 36-7-24-4 & = & 1 \\ 29-24-4 & = & 1 \\ 5-4 & = & 1 \\ 1 & = & 1 \quad \checkmark \end{array}[/latex] The solution checks.

	[latex]2(3k-1)-5k=-2-7[/latex]
Distribute on the left, subtract on the right.	[latex]6k-2-5k=-9[/latex]
Use the Commutative Property of Addition.	[latex]6k-5k-2=-9[/latex]
Combine like terms.	[latex]k-2=-9[/latex]
Undo subtraction by using the Addition Property of Equality.	[latex]k-2(\color{red}{+2}\color{black})=-9(\color{red}{+2}\color{black})[/latex]
Simplify.	[latex]k=-7[/latex]
Check.	Let [latex]k=-7[/latex]. [latex]\begin{array}{rcl} 2(3k-1)-5k & = & -2-7 \\ 2\left(3(\color{red}{-7}\color{black})-1\right)-5(\color{red}{-7}\color{black}) & = & -2-7 \\ 2(-21-1)-5(-7) & = & -9 \\ 2(-22)+35 & = & -9 \\ -44+35 & = & -9 \\ -9 & = & -9 \quad \checkmark \end{array}[/latex] The solution checks.