How to Solve Linear Equations?

Linear equations are fundamental in algebra and are equations that make a straight line when graphed. They typically look like this: $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Steps to Solve Linear Equations

1. Simplify Both Sides
   First, simplify both sides of the equation if necessary. This involves combining like terms and eliminating any parentheses. For example:

   $2(x + 3) = 4x – 1$

   Distribute the 2:

   $2x + 6 = 4x – 1$

2. Move Variable Terms to One Side
   Next, get all the variable terms on one side of the equation and the constant terms on the other. You can do this by adding or subtracting terms from both sides. For instance:

   $2x + 6 = 4x – 1$

   Subtract $2x$ from both sides:

   $6 = 2x – 1$

3. Isolate the Variable
   Now, isolate the variable by performing the inverse operation. In this case, add 1 to both sides:

   $6 + 1 = 2x$

   $7 = 2x$

   Then divide both sides by 2:

   $x = frac{7}{2}$

4. Check Your Solution
   Always plug your solution back into the original equation to ensure it is correct. For our example:

   $2big(frac{7}{2} + 3big) = 4big(frac{7}{2}big) – 1$

   Simplify both sides:

   $2big(frac{7}{2} + frac{6}{2}big) = 4big(frac{7}{2}big) – 1$

   $2big(frac{13}{2}big) = 2 times 7 – 1$

   $13 = 13$

   The solution $x = frac{7}{2}$ is correct.

Example Problem

Let’s solve another example for practice:

$3x – 5 = 2x + 4$

1. Simplify Both Sides
   The equation is already simplified.

2. Move Variable Terms to One Side
   Subtract $2x$ from both sides:

   $3x – 2x – 5 = 4$

   $x – 5 = 4$

3. Isolate the Variable
   Add 5 to both sides:

   $x = 9$

4. Check Your Solution
   Plug $x = 9$ back into the original equation:

   $3(9) – 5 = 2(9) + 4$

   $27 – 5 = 18 + 4$

   $22 = 22$

   The solution $x = 9$ is correct.

Conclusion

Solving linear equations involves simplifying both sides, moving variable terms to one side, isolating the variable, and checking your solution. With practice, these steps will become second nature.

Citations

1. 1. Khan Academy – Solving Linear Equations
2. 2. Math is Fun – Linear Equations
3. 3. Purplemath – Solving Linear Equations