Precalculus Study Guide: Algebraic Expressions, Linear Equations, and Graphing Lines

Algebraic Expressions and Linear Equations

Computing Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. Understanding how to compute and simplify these expressions is foundational for all further study in algebra and precalculus.

• Definition: An algebraic expression is a combination of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division).

• Simplifying Expressions: Combine like terms and use the distributive property to simplify expressions.

• Evaluating Expressions: Substitute given values for variables and perform the indicated operations.

• Example: Simplify $2x + 3x - 4$.

  • Combine like terms: $2x + 3x = 5x$

  • Result: $5x - 4$

Solving Linear Equations and Inequalities in One Variable

Linear equations and inequalities in one variable are fundamental tools for solving real-world problems and for building more advanced mathematical concepts.

• Linear Equation: An equation of the form $ax + b = 0$, where $a \neq 0$.

• Solution: Isolate the variable to find its value.

  • Example: Solve $3x - 7 = 5$.

    • Add 7 to both sides: $3x = 12$

    • Divide by 3: $x = 4$

• Linear Inequality: An inequality of the form $ax + b \lt c$ or $ax + b \gt c$.

• Solution: Use similar steps as equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

  • Example: Solve $2x - 3 \leq 7$.

    • Add 3: $2x \leq 10$

    • Divide by 2: $x \leq 5$

Linear Equations in Two Variables and Graphing Lines

Manipulating Two-Variable Linear Equations

Linear equations in two variables describe straight lines in the coordinate plane. Manipulating these equations allows us to analyze and graph lines effectively.

• Standard Form: $Ax + By = C$

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

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

• Converting Between Forms: Use algebraic manipulation to rewrite equations as needed.

• Example: Convert $2x + 3y = 6$ to slope-intercept form.

  • Subtract $2x$: $3y = -2x + 6$

  • Divide by 3: $y = -\frac{2}{3}x + 2$

Graphing Lines

Graphing linear equations helps visualize relationships between variables and solve systems of equations.

• Using Slope and Intercept: Start at the y-intercept $(0, b)$ and use the slope $m$ to find another point.

• Plotting Points: Choose values for $x$, solve for $y$, and plot the resulting points.

• Example: Graph $y = 2x - 1$.

  • Y-intercept: $(0, -1)$

  • Slope: $2$ (rise 2, run 1)

  • Plot $(0, -1)$ and $(1, 1)$, then draw the line through these points.

Standard Forms of Equations of Lines

Understanding the different forms of linear equations allows for flexibility in solving and graphing problems.

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$

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

• Horizontal Line: $y = k$ (slope $m = 0$)

• Vertical Line: $x = h$ (undefined slope)

• Example: Write the equation of a line passing through $(2, 3)$ with slope $-4$.

  • Point-slope form: $y - 3 = -4(x - 2)$

  • Simplify: $y = -4x + 8 + 3 = -4x + 11$

Finding Intersections of Lines Using Substitution

The intersection point of two lines is the solution to their system of equations. The substitution method is a common algebraic technique for finding this point.

• Substitution Method:

  1. Solve one equation for one variable.

  2. Substitute this expression into the other equation.

  3. Solve for the remaining variable.

  4. Substitute back to find the other variable.

• Example: Find the intersection of $y = 2x + 1$ and $x + y = 7$.

  • Substitute $y$ from the first equation into the second: $x + (2x + 1) = 7$

  • Simplify: $3x + 1 = 7$

  • $3x = 6 \implies x = 2$

  • Substitute $x = 2$ into $y = 2x + 1$: $y = 2(2) + 1 = 5$

  • Intersection point: $(2, 5)$

Linear Models

Creating and Using Linear Models

Linear models are equations that represent real-world relationships with a constant rate of change. They are widely used in science, business, and everyday problem-solving.

• Definition: A linear model is an equation of the form $y = mx + b$ that describes a linear relationship between two variables.

• Steps to Create a Linear Model:

  1. Identify the variables and collect data.

  2. Determine the slope $m$ (rate of change) and y-intercept $b$ (initial value).

  3. Write the equation in slope-intercept form.

• Example: A car rental company charges a $50$dollar base fee plus $20$ dollars per day. The cost $C$ for $d$ days is $C = 20d + 50$.

Additional info: These notes are based on a course outline referencing textbook sections R7, 1.1, 1.5, 1.7, 2.2, and 2.3, which correspond to review of algebraic expressions, solving linear equations and inequalities, manipulating and graphing linear equations, and creating linear models. The content has been expanded to provide a self-contained study guide for Precalculus students.