14.3 Linear Functions, Slope-Intercept Form & Systems of Linear Equations

Linear Functions, Slope-Intercept Form & Systems of Linear Equations

Quick Answer: On the WEST-B Mathematics subtest (Objective 0017), linear functions and systems questions test your coordinate algebra and multi-variable problem-solving. Remember: A relation is a function if every x has exactly one y (passes the Vertical Line Test). Slope is m = (y_2 - y_1)/(x_2 - x_1). Horizontal lines have m = 0 (y = c); vertical lines have undefined slope (x = c). Parallel lines have equal slopes (m_1 = m_2); perpendicular lines have negative reciprocal slopes (m_2 = -1/m_1). For a 2 × 2 system of linear equations, use Substitution when a variable has coefficient ±1, and Elimination when coefficients align nicely. A system has 1 solution (intersecting lines), no solution (parallel lines), or infinite solutions (identical lines).

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1. Relations, Functions, Domain, Range & Function Notation

An algebraic relation is any set of ordered pairs (x, y). A function is a specialized relation where each input value x in the domain is paired with exactly one unique output value y in the range.

+---------------------------------------------------------------------------------------------------+
|                                 FUNCTION DETERMINATION CRITERIA                                   |
|                                                                                                   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | REPRESENTATION    | IS A FUNCTION (YES)        | NOT A FUNCTION (NO)                      |   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Set of Pairs      | {(1, 4), (2, 7), (3, 4)}   | {(1, 4), (1, 9), (2, 7)}                 |   |
|   |                   | (Outputs can repeat!)      | (Input 1 has two different outputs: 4 & 9|   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Mapping Diagram   | Each x has exactly 1 arrow | An x has 2 or more arrows departing      |   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Graphical (VLT)   | Any vertical line cuts     | A vertical line intersects the curve at  |   |
|   |                   | the curve at AT MOST 1 pt  | TWO or more points (fails VLT)           |   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Equation Form     | y = 3x - 5, y = x² + 2     | x² + y² = 25 (Circle), x = 4 (Vertical)  |   |
|   +-------------------+----------------------------+------------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Function Notation: f(x)

The notation f(x) (read "f of x") represents the output value y generated when input x is processed by function f.

• Evaluating Functions: If f(x) = -2x² + 5x - 3, evaluate f(-3): f(-3) = -2(-3)² + 5(-3) - 3 = -2(9) - 15 - 3 = -18 - 15 - 3 = -36
• Solving for Input: If g(x) = 4x - 7, find x when g(x) = 25: 4x - 7 = 25 => 4x = 32 => x = 8

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2. Rate of Change & Slope (m)

The slope (m) of a non-vertical line passing through points (x_1, y_1) and (x_2, y_2) is the ratio of vertical change (rise, Δy) to horizontal change (run, Δx): m = Rise / Run = Δy / Δx = (y_2 - y_1) / (x_2 - x_1)

+---------------------------------------------------------------------------------------------------+
|                                    THE FOUR TYPES OF SLOPE                                        |
|                                                                                                   |
|        Positive Slope (m > 0)                       Negative Slope (m < 0)                        |
|               /                                            \                                      |
|              /   Rises left-to-right                        \   Falls left-to-right               |
|             /    Δy > 0 as Δx > 0                            \  Δy < 0 as Δx > 0                  |
|            /                                                  \                                   |
|                                                                                                   |
|        Zero Slope (m = 0)                           Undefined Slope (m = undefined)               |
|       ------------------- Horizontal Line                    |   Vertical Line                    |
|                           Equation: y = c                    |   Equation: x = c                  |
|                           Δy = 0, Δx ≠ 0                     |   Δx = 0 (division by zero!)       |
|                           Function? YES                      |   Function? NO                     |
+---------------------------------------------------------------------------------------------------+

Trap Alert: A horizontal line has slope m = 0 and equation y = c. A vertical line has undefined slope and equation x = c. Vertical lines are NOT functions because they fail the Vertical Line Test infinitely.

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3. Forms of Linear Equations & Intercepts

Linear equations on the Cartesian coordinate plane can be written in three interchangeable algebraic forms.

+---------------------------------------------------------------------------------------------------+
|                                 FORMS OF LINEAR EQUATIONS MATRIX                                  |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | FORM NAME             | STANDARD FORMULA            | KEY PARAMETERS & UTILITY            |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Slope-Intercept Form  | y = mx + b                  | m = Slope                           |   |
|   |                       |                             | b = y-intercept at point (0, b)     |   |
|   |                       |                             | (Ideal for graphing & slope reading)|   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Point-Slope Form      | y - y₁ = m(x - x₁)          | m = Slope                           |   |
|   |                       |                             | (x₁, y₁) = Any known point on line  |   |
|   |                       |                             | (Ideal for writing line equations)  |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Standard Form         | Ax + By = C                 | A, B, C are integers, A ≥ 0         |   |
|   |                       |                             | Slope m = -A/B                      |   |
|   |                       |                             | x-intercept = (C/A, 0)              |   |
|   |                       |                             | y-intercept = (0, C/B)              |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Finding Intercepts

• x-intercept: The point where the graph crosses the x-axis. Set y = 0 and solve for x.
• y-intercept: The point where the graph crosses the y-axis. Set x = 0 and solve for y.
• Example: For 3x - 5y = 30:
  • x-intercept: 3x - 5(0) = 30 => 3x = 30 => x = 10 => (10, 0)
  • y-intercept: 3(0) - 5y = 30 => -5y = 30 => y = -6 => (0, -6)

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4. Parallel & Perpendicular Lines

The geometric relationship between two lines is determined entirely by their slopes.

+---------------------------------------------------------------------------------------------------+
|                             PARALLEL VS. PERPENDICULAR LINE CRITERIA                              |
|                                                                                                   |
|   +-------------------+-----------------------------+-----------------------------------------+   |
|   | RELATIONSHIP      | SLOPE CONDITION             | VISUAL / GEOMETRIC EFFECT               |   |
|   +-------------------+----------------------------+-----------------------------------------+   |
|   | Parallel Lines    | m₁ = m₂                     | Same steepness, never intersect         |   |
|   | (L₁ ∥ L₂)         | (b₁ ≠ b₂)                   | (Equal slopes, distinct y-intercepts)   |   |
|   +-------------------+-----------------------------+-----------------------------------------+   |
|   | Perpendicular     | m₁ · m₂ = -1                | Intersect at right angles (90°)         |   |
|   | Lines (L₁ ⊥ L₂)   | m₂ = -1 / m₁                | Slopes are OPPOSITE RECIPROCALS         |   |
|   +-------------------+-----------------------------+-----------------------------------------+   |
|   | Horizontal /      | m_horiz = 0                 | Always perpendicular to each other      |   |
|   | Vertical Lines    | m_vert = undefined          | y = a  is perpendicular to  x = b       |   |
|   +-------------------+-----------------------------+-----------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Finding Opposite Reciprocals

• If m_1 = 2/5 => m_2 = -5/2
• If m_1 = -4 => m_2 = +1/4
• If m_1 = -7/3 => m_2 = +3/7

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5. Systems of Linear Equations in Two Variables

A system of linear equations consists of two or more linear equations containing the same variables. Equation 1: a₁x + b₁y = c₁ Equation 2: a₂x + b₂y = c₂

+---------------------------------------------------------------------------------------------------+
|                              SYSTEM CLASSIFICATION & SOLUTIONS                                    |
|                                                                                                   |
|   +-----------------------+-----------------------+----------------------+--------------------+   |
|   | CLASSIFICATION        | SLOPES & INTERCEPTS   | GEOMETRIC GRAPH      | NUMBER OF SOLUTIONS|   |
|   +-----------------------+-----------------------+----------------------+--------------------+   |
|   | Consistent &          | Slopes are DIFFERENT  | Two lines intersect  | Exactly ONE        |   |
|   | Independent           | (m₁ ≠ m₂)             | at a single point    | unique pair (x, y) |   |
|   +-----------------------+-----------------------+----------------------+--------------------+   |
|   | Inconsistent          | Slopes are EQUAL,     | Parallel lines       | NO SOLUTION        |   |
|   |                       | Intercepts DIFFERENT  | (never intersect)    | ∅ or { }           |   |
|   |                       | (m₁ = m₂, b₁ ≠ b₂)    |                      |                    |   |
|   +-----------------------+-----------------------+----------------------+--------------------+   |
|   | Consistent &          | Slopes are EQUAL,     | Coincident lines     | INFINITELY MANY    |   |
|   | Dependent             | Intercepts EQUAL      | (same line graphed)  | solutions          |   |
|   |                       | (m₁ = m₂, b₁ = b₂)    |                      |                    |   |
|   +-----------------------+-----------------------+----------------------+--------------------+   |
+---------------------------------------------------------------------------------------------------+

Algebraic Methods for Solving Systems

Method 1: Substitution

Best when at least one variable has a coefficient of 1 or -1.

1. Isolate one variable in one equation (e.g., y = 3x - 4).
2. Substitute that expression into the other equation.
3. Solve the resulting single-variable equation.
4. Back-substitute to find the second variable.

Method 2: Elimination / Linear Combination

Best when equations are in standard form Ax + By = C with non-unit coefficients.

1. Multiply one or both equations by chosen non-zero constants so that the coefficients of one variable become exact opposites (e.g., +6y and -6y).
2. Add the two equations vertically to eliminate that variable.
3. Solve for the remaining variable.
4. Back-substitute the value into either original equation to find the other variable.

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6. Real-World Systems Modeling & Applications

WEST-B questions frequently present multi-variable real-world scenarios requiring algebraic systems modeling.

1. Ticket Sales / Quantity-Value Problems

• Quantity Equation: Count of items (x + y = Total Items)
• Value / Revenue Equation: Monetary worth ((Price₁)x + (Price₂)y = Total Revenue)

2. Break-Even Analysis

• Cost Function: C(x) = Fixed Cost + (Variable Cost per unit)x
• Revenue Function: R(x) = (Selling Price per unit)x
• Break-Even Point: Set R(x) = C(x) and solve for x.

3. Mixture and Solution Problems

• Total Volume Equation: v₁ + v₂ = v_total
• Pure Substance Equation: c₁v₁ + c₂v₂ = (c_final)(v_total)

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7. Step-by-Step Worked Problems & Exact Derivations

Problem 1: Equation of a Perpendicular Line

Problem: Write the equation in slope-intercept form (y = mx + b) for the line that passes through the point (-3, 4) and is perpendicular to the line 2x - 6y = 15.

Step-by-Step Solution:

1. Find the slope of the given line: Convert 2x - 6y = 15 to slope-intercept form: -6y = -2x + 15 => y = (-2/-6)x + (15/-6) => y = (1/3)x - 5/2 The slope of the given line is m₁ = 1/3.
2. Determine the perpendicular slope (m₂): Perpendicular lines have negative reciprocal slopes: m₂ = -1 / m₁ = -1 / (1/3) = -3
3. Use point-slope form with m = -3 and point (x₁, y₁) = (-3, 4): y - y₁ = m(x - x₁) y - 4 = -3(x - (-3)) y - 4 = -3(x + 3)
4. Distribute and simplify to slope-intercept form: y - 4 = -3x - 9 y = -3x - 5

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Problem 2: Solving a 2 × 2 System by Elimination

Problem: Solve the system of linear equations: 4x + 3y = -1 3x - 2y = 12 Determine the solution (x, y) and calculate the value of x + y.

Step-by-Step Solution:

1. Choose variable to eliminate: Eliminate y by finding the LCM of 3 and 2, which is 6.
2. Multiply Equation 1 by 2 and Equation 2 by 3: 2(4x + 3y) = 2(-1) => 8x + 6y = -2 3(3x - 2y) = 3(12) => 9x - 6y = 36
3. Add the two equations vertically to eliminate y: (8x + 9x) + (6y - 6y) = -2 + 36 17x = 34 => x = 2
4. Substitute x = 2 into Equation 1 to find y: 4(2) + 3y = -1 8 + 3y = -1 3y = -9 => y = -3
5. State solution and calculate required value: Solution: (x, y) = (2, -3) x + y = 2 + (-3) = -1

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Problem 3: Real-World Ticket Sales Word Problem

Problem: A school auditorium sold 350 total tickets for a musical performance, generating a total revenue of $3,850. Adult tickets cost $14 each and student tickets cost $7 each. How many adult tickets were sold?

Step-by-Step Solution:

1. Define the variables: Let A = number of adult tickets, S = number of student tickets.
2. Construct the system of equations:
   • Ticket count: A + S = 350
   • Revenue: 14A + 7S = 3850
3. Use substitution: Express S in terms of A: S = 350 - A
4. Substitute into revenue equation: 14A + 7(350 - A) = 3850 14A + 2450 - 7A = 3850
5. Combine like terms and solve for A: 7A + 2450 = 3850 7A = 1400 => A = 200
6. Calculate student tickets and verify: S = 350 - 200 = 150 Check revenue: 14(200) + 7(150) = 2800 + 1050 = 3850. Exact match! Therefore, 200 adult tickets were sold.