Linear Equations

Linear Equations

Linear equations are the foundation of algebra and represent straight lines when graphed. Here's what you need to know:

Standard Forms

Slope-Intercept Form

$$y = mx + b$$

Where:

• $$m$$ is the slope (rate of change)
• $$b$$ is the y-intercept (where the line crosses the y-axis)

Example: $$y = 2x + 3$$ has slope 2 and y-intercept 3

Point-Slope Form

$$y - y_1 = m(x - x_1)$$

Where:

• $$m$$ is the slope
• $$(x_1, y_1)$$ is a point on the line

Example: A line with slope 4 passing through (2, 5) has equation $$y - 5 = 4(x - 2)$$

Standard Form

$$Ax + By = C$$

Where A, B, and C are constants (usually integers) and A ≥ 0

Example: $$3x + 2y = 12$$

Solving Linear Equations

Basic Principles

• What you do to one side of the equation, you must do to the other side
• Goal: Isolate the variable

Steps to Solve

1. Simplify each side (combine like terms)
2. Move all variable terms to one side
3. Move all constant terms to the other side
4. Divide both sides by the coefficient of the variable

Example: Solve $$3x + 5 = 2x - 7$$

Step 1: Already simplified

Step 2: $$3x - 2x = -7 - 5$$

Step 3: $$x = -12$$

Equations with Fractions

Multiply all terms by the least common multiple (LCM) of the denominators to eliminate fractions.

Example: Solve $$\frac{x}{3} + \frac{2x}{5} = 4$$

Multiply by LCM of 3 and 5, which is 15:

$$15 \cdot (\frac{x}{3} + \frac{2x}{5} = 4)$$

$$5x + 6x = 60$$

$$11x = 60$$

$$x = \frac{60}{11}$$

Applications of Linear Equations

Word Problems

Steps to solve word problems:

1. Identify the unknown quantity and assign a variable
2. Express all other quantities in terms of this variable
3. Form an equation based on the given conditions
4. Solve the equation
5. Check your answer in the context of the problem

Common Applications

Motion problems: distance = rate × time

Example: If a car travels at 60 mph, how long will it take to travel 150 miles?

time = distance ÷ rate = 150 ÷ 60 = 2.5 hours

Mixture problems: amount × concentration = quantity of substance

Example: How much 20% salt solution should be mixed with 50 ml of 50% salt solution to get a 30% salt solution?

Let x = amount of 20% solution

0.20x + 0.50(50) = 0.30(x + 50)

0.20x + 25 = 0.30x + 15

25 - 15 = 0.30x - 0.20x

10 = 0.10x

x = 100 ml

Graphing Linear Equations

Methods for Graphing

1. Using slope and y-intercept: Plot the y-intercept, then use the slope to find additional points
2. Using x and y intercepts: Find where the line crosses each axis
3. Using a table of values: Choose x-values, calculate corresponding y-values

Finding the Equation from a Graph

1. Identify two points on the line
2. Calculate the slope: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
3. Use point-slope form to write the equation
4. Convert to desired form

Common Pitfalls and Tips

• Watch for sign errors when moving terms between sides
• Remember that dividing by a negative number reverses the inequality sign
• Check your solution by substituting back into the original equation
• Horizontal lines have equation y = c (slope = 0)
• Vertical lines have equation x = c (undefined slope)
• Parallel lines have the same slope
• Perpendicular lines have slopes that are negative reciprocals of each other (product = -1)