One fundamental topic in any Algebra 1 course is the process of solving one variable equations. The key concept is to work backwards from the order of operations being applied to the equation and isolate the variable to get a solution. The best approach is to jump in and solve problems, so here we go!
Solve the two-step equation: \(-3b-2=-41\).
The original equation has the following mathematical operations: the variable \(b\) multiplied by \(-3\), then followed by subtracting \(2\) and is set equal to \(-41\). The steps to solve are to reverse these operations: add \(2\) first to BOTH sides, since adding reverses subtracting. Followed by dividing by \(-3\), since divide reverses multiply, and that will isolate the variable \(b\) giving a potential solution.
| Add \(2\) to both sides: | \(-3b-2\color{blue}{+2}=-41\color{blue}{+2}\) |
| \(\implies-3b=-39\) | |
| Divide \(-3\) on both sides: | \(\large{\frac{-3b}{\color{blue}{-3}}=\frac{-39}{\color{blue}{-3}}}\) |
| \(\implies b=13\;\color{green}{\checkmark}\) |
Solve the equation: \(2(5+7w)+6w=-130.\)
| Distribute the \(2\): | \(10+14w+6w=-130\) |
| Combine like terms: | \(10+20w=-130\) |
| Solve two-step: | \(20w=-130-10\) |
| \(20w=-140\) | |
| \(w=\large{\frac{-140}{20}}\) | |
| \(w=-7\;\color{green}{\checkmark}\) |
This equation has a multiplier of \(2\) that needs the Distributive Property applied first. The result gives two variable terms on the left side, which must be combined into a single variable term: \(14w+6w=20w\). Now the equation is another two-step equation much like the first example above. The math operations here are multiply by \(20\), then add \(10\) to get \(-130\). The reversal procedure is to subtract \(10\), then divide by \(20\) to isolate the variable \(w\).
Solve the equation: \(-5(6+x)=-15-2x\).
This equation has the variable \(x\) on both sides of the equation. Notice also, there is another Distributive Property on the left side. Start by distributing the \(-5\), be careful, it is negative, and combine any like terms. The most complicated step is then to "move" all variable terms to the same side of the equation. It does not matter what side terms are moved to, just reverse the math operation being applied. In this example, the variable term on the left of \(5x\) is subtracted, so reversal means to add \(5x\) to BOTH sides of the equation. Once variables are now on the same side, the result is a two-step equation solved as above.
| Distribute \(-5\): | \(\implies -30-5x=-15-2x\) |
| Move variables to one side: | \(\implies -30-5x\color{blue}{+5x}=-15-2x\color{blue}{+5x}\) |
| Combine like terms: | \(\implies -30=-15+3x\) |
| Solve two-step: | \(\implies -30+15=3x\implies -15=3x\) |
| \(\implies \large{\frac{-15}{3}}\normalsize{=x}\) | |
| \(\implies -5=x\;\color{green}{\checkmark}\) |
Solve the equation: \(4y+1+4y=1-8y+6y\).
| Combine like terms: | \(\implies 8y+1=1-2y\) |
| Variables on Both Sides: | \(\implies 8y\color{blue}{+2y}=1\color{blue}{+2y}\implies 10y+1=1\) |
| Solve two-step: | \(\implies 10y+1\color{blue}{-1}=1\color{blue}{-1}\implies 10y=0\) |
| \(\implies y=\large{\frac{0}{10}}\normalsize{\implies y=0\;\color{green}{\checkmark}}\) |
Sometimes in the solving process, the variables may cancel out leaving some strange results. If the statement is false, then the equation has no solution. If the statement is true, then all real numbers work in the equation. Consider the following examples:
| \(\implies6x+11=6x-5\) |
| \(\implies6x\color{blue}{-6x}+11=6x\color{blue}{-6x}-5\) |
| \(\implies11=-5\) |
| FALSE: equation has no solution |
| \(\implies2x-6+7=9+2x-8\) |
| \(\implies2x+1=2x+1\) |
| \(\implies2x\color{blue}{-2x}+1=2x\color{blue}{-2x}+1\) |
| \(\implies1=1\) |
| TRUE: all real numbers are solutions |
Final thoughts about solving equations. Once the variable has been isolated, like \(x=3\), a quick check can determine if the value is a solution by plugging this value back into the original equation. Like above, a TRUE statement implies the value is in fact a solution, and a FALSE statement implies the value is NOT a solution. While checking solutions has not been shown here, it is good practice to determine if any equation has been solved correctly.