College Algebra: Linear Equations, Inequalities, Graphs, and Systems – Study Guide

Linear Equations & Applications

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable(s) appear only to the first power. Solving these equations involves isolating the variable using algebraic operations.

• Definition: A linear equation in one variable has the form $ax + b = c$.

• Steps: Combine like terms, isolate the variable, and solve.

• Example: Solve $3(2x - 1) = 2(3x - 7)$ $6x - 3 = 6x - 14$ $-3 = -14$ (No solution; contradiction)

• Applications: Word problems may require setting up equations from context, such as cost models or ticket sales.

Applications of Linear Equations

Real-world problems often require translating a scenario into a linear equation and solving for unknowns.

• Example: A rideshare charges $3.50 base + $1.20 per mile. Write$C(m)$and compute$C(12)$.$C(m) = 3.50 + 1.20mC(12) = 3.50 + 1.20 \times 12 = 17.90$

Inequalities

Solving Linear Inequalities

Linear inequalities are similar to equations but use inequality symbols (<, >, ≤, ≥). Solutions are often expressed as intervals or in set-builder notation.

• Definition: A linear inequality has the form $ax + b < c$ or $ax + b \leq c$.

• Steps: Solve as with equations, but reverse the inequality sign when multiplying/dividing by a negative.

• Example: Solve $2x - 4 \leq 16$ $2x \leq 20$ $x \leq 10$ Set-builder: $\{x \mid x \leq 10\}$ Interval: $(-\infty, 10]$

Absolute Value Equations

Solving Absolute Value Equations

Absolute value equations require considering both the positive and negative cases of the expression inside the absolute value.

• Definition: $|A| = B$ means $A = B$ or $A = -B$ (if $B \geq 0$).

• Example: Solve $|x - 4| = 8$ $x - 4 = 8$ or $x - 4 = -8$ $x = 12$ or $x = -4$

Graphs of Linear Equations

Graphing and Interpreting Linear Equations

Linear equations can be graphed as straight lines. Key features include the slope and y-intercept.

• General Form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

• Finding Intercepts: Set $x = 0$ for y-intercept, $y = 0$ for x-intercept.

• Example: $y = \frac{2}{3}x - 3$ y-intercept: $-3$ Slope: $\frac{2}{3}$

Parallel & Perpendicular Lines

Identifying and Writing Equations

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: $m_1 = m_2$

• Perpendicular: $m_1 \cdot m_2 = -1$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Example: Find the equation of a line through $(4, -1)$ parallel to $y = -2x + 7$ Slope $m = -2$ $y + 1 = -2(x - 4)$ $y = -2x + 8 - 1$ $y = -2x + 7$

Functions

Evaluating and Combining Functions

Functions assign each input exactly one output. They can be evaluated and combined using algebraic operations.

• Definition: $f(x)$ is a function of $x$.

• Operations: $f(x) + g(x)$, $f(x) - g(x)$, $f(x) \cdot g(x)$, $f(x) / g(x)$

• Example: If $f(x) = x^2 - 4x + 6$, $g(x) = 5x - 2$, find $f(1) + g(-1)$ $f(1) = 1 - 4 + 6 = 3$ $g(-1) = -5 - 2 = -7$ $f(1) + g(-1) = 3 + (-7) = -4$

Systems of Linear Equations

Solving Systems by Substitution and Elimination

Systems of equations involve finding values that satisfy all equations simultaneously. Common methods are substitution and elimination.

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination: Add/subtract equations to eliminate a variable.

• Example: Solve $\begin{cases} 3x + 2y = 11 \\ x - y = 9 \end{cases}$ From second: $x = y + 9$ Substitute: $3(y + 9) + 2y = 11$ $3y + 27 + 2y = 11$ $5y = -16$ $y = -3.2$ $x = -3.2 + 9 = 5.8$

Applications with Systems

Word Problems Involving Systems

Many real-world problems require setting up and solving systems of equations to find unknown quantities.

• Example: A bookstore sells notebooks for $2 and binders for $6. Sold 120 items for $420. How many of each? Let$x$= notebooks,$y$= binders.$x + y = 120$$2x + 6y = 420$ Solve: $x = 90$, $y = 30$

Summary Table: Key Concepts

Concept	Key Formula/Property
Lines	$y = mx + b$, $m = \frac{\Delta y}{\Delta x}$, Point-slope: $y - y_1 = m(x - x_1)$
Parallel	Same slope: $m_1 = m_2$
Perpendicular	Negative reciprocal slopes: $m_1 \cdot m_2 = -1$
Absolute Value	$|A| = B \implies A = B$ or $A = -B$ ($B \geq 0$)
Inequalities	Multiply/divide by negative: flip the sign
Functions	$f(x) + g(x)$, $f(x) - g(x)$, $f(x) \cdot g(x)$, $f(x) / g(x)$
Systems	Substitution or elimination; always verify in both equations

Additional info:

• Practice problems mirror midterm exam scope and difficulty.

• Always check solutions for reasonableness and verify in original equations.

• Set-builder and interval notation are standard for expressing solution sets.