Linear Equations in One Variable

A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable. In Grade 9, these equations become more complex, often featuring variables on both sides, parentheses, or unknown coefficients.

The Standard Solving Process

To solve a linear equation, your main goal is to isolate the variable on one side of the equals sign. Follow these steps:

1. Distribute to remove any parentheses.
2. Combine like terms on each side of the equation.
3. Move variables to one side (using addition or subtraction).
4. Isolate the variable by multiplying or dividing.

Example: Solve $3(x - 2) = 2x + 5$

First, distribute the $3$: $3x - 6 = 2x + 5$

Next, subtract $2x$ from both sides to get all $x$'s on the left: $x - 6 = 5$

Finally, add $6$ to both sides: $x = 11$

Equations with Unknown Coefficients

Sometimes, an equation includes other letters (parameters) instead of numbers. You treat these exactly like numbers and factor out the variable to isolate it.

Example: Solve $ax + 3 = bx - 1$ for $x$

Move all terms containing $x$ to one side, and constants to the other. Subtract $bx$ and subtract $3$: $ax - bx = -1 - 3$ $ax - bx = -4$

Now, factor out the $x$ on the left side: $x(a - b) = -4$

Divide both sides by $(a - b)$ to isolate $x$ (assuming $a \neq b$): $x = \frac{-4}{a - b}$

Number of Solutions

Not all linear equations have exactly one solution. By simplifying an equation, you can determine how many solutions it has:

• One Solution: The equation simplifies to $x = \text{number}$. The lines intersect at one point.
• No Solution: The equation simplifies to a false statement, like $3 = 5$. This means no value of $x$ will ever make the equation true.
• Infinitely Many Solutions: The equation simplifies to a true statement, like $4 = 4$ or $x = x$. This means any real number is a valid solution.

Example: Determine the number of solutions for $4(x + 1) = 4x + 4$

Distribute the $4$ on the left side: $4x + 4 = 4x + 4$

Subtract $4x$ from both sides: $4 = 4$

Because $4 = 4$ is always a true statement, regardless of what $x$ is, this equation has infinitely many solutions.