Methods of Solving Equations With Variables
- If an equation has a single variable, designated by a letter 'x', the known numbers in the equation can be moved to one side of the equal sign in order to solve for the unknown letter 'x' variable. For example, the equation 5+x=9 can be rewritten as an equivalent equation 5-5+x=9-5. In this equation, the number five was subtracted from both sides of the equal sign. Since 5-5=0, this equivalent equation can be simplified to x=9-5, and x=4. This is an example of solving an equation for a single variable.
- If an equation contains multiple appearances of the same variable, the equation can be modified to simplify the equation into one appearance of the variable, and then the equation solves for this single variable, as demonstrated in the above step. For example, the equation 2x+4x=24 can be simplified into 6x=24, and then reduced to 6x/6 = 24/6, or x=4.
- Algebraic equations become more complicated when two or more variables appear in multiple equations. Two equations, each containing x and y variables, can be combined using a method called substitution in order to solve for both of the variables. For example, this method can be used to solve for the variables "x" and "y" in the following equations, 2x+3y=48 and 3x+2y=12. Using single variable simplification, the first equation can be rewritten as x=(48-3y)/2. Now that we have solved for x in terms of the y variable, the solution for x can be plugged into the second equation, and we can solve for y, and rewrite in the second equation as 3[(48-3y)/2] +6y=12. This equation simplifies into 9y=120, or y=40/3. Now the solution for the y variable can be plugged back into the first equation and solved for the x variable. If y=40/3, then 2x+3(40/3)=48, or 2x+40=48, or x=4.
- Multiple equations with more than one variable each can be added together in order to eliminate one of the variables, and solved for the remaining single variable. Given a first equation 6x+9y=72, and the second equation 6x+3y=36, these two equations can be stacked on top of each other, and one of the equations subtracted from the other as if they were real numbers. If the second equation is subtracted from the first, we have (6x-6x)+(9y-3y)=72-36, or 6y=36, so y= 6. By plugging this number into either of the two equations, we can solve for x. 6x+3*6=36, or 6x=18, or x=3.
