Welcome back to Professor Baker's Math Class! Today, we are tackling one of the most fundamental skills in algebra: Solving Linear Equations. Whether you are dealing with a simple two-step problem or a long equation with parentheses on both sides, the process remains the same. By following a structured approach, you can solve for $x$ with confidence every time.

The 5 Steps to Success

As outlined in our class notes, there is a logical order of operations to follow when isolating a variable. Memorize these five steps:

  1. Remove Parentheses: Use the Distributive Property to clear any grouping symbols.
  2. Combine Like Terms: Simplify expressions on the left and right sides of the equal sign independently.
  3. Move Variables to One Side: If $x$ appears on both sides, use addition or subtraction to move them together.
  4. Remove Addition and Subtraction: Isolate the variable term by moving constants to the other side.
  5. Remove Multiplication and Division: Solve for the single variable (get $x$ by itself).

Example: Putting it all together

Let's look at a complex example from the class notes (Page 2) that uses all five steps. Don't let the length of the equation scare you; just take it one step at a time.

The Equation:

$$7(x+4) - 3x + 5 = 2x - 4 + 3$$

Step 1: Remove Parentheses
Distribute the $7$ into $(x+4)$:

$$7x + 28 - 3x + 5 = 2x - 4 + 3$$

Step 2: Combine Like Terms
On the left side, combine $7x$ and $-3x$, and combine $28$ and $5$. On the right side, combine $-4$ and $3$:

$$4x + 33 = 2x - 1$$

Step 3: Move Variables to One Side
Let's move the $2x$ from the right to the left by subtracting $2x$ from both sides:

$$2x + 33 = -1$$

Step 4: Remove Addition and Subtraction
Now, get the $2x$ by itself by subtracting $33$ from both sides:

$$2x = -34$$

Step 5: Remove Multiplication and Division
Finally, divide by $2$ to isolate $x$:

$$x = -17$$

Working with Fractions

Sometimes you will encounter coefficients that are fractions. As seen in the notes (Page 4), you have two ways to handle an equation like $\frac{2}{3}x = 10$.

  • Method 1: Multiply by the denominator ($3$) first to get $2x = 30$, then divide by the numerator ($2$).
  • Method 2 (The Reciprocal): Multiply both sides by the reciprocal of the fraction.
$$\left(\frac{3}{2}\right) \cdot \frac{2}{3}x = 10 \cdot \left(\frac{3}{2}\right)$$ $$x = 15$$

Both methods yield the same result! Remember to check the attached PDF for more examples, including simple one-step equations and more practice with fractions.