RC 3 D1

​RC 3:
Writing and Solving Linear Functions,

Equations, and Inequalities

By Kenneth Snyder

Domain and Range

• Domain (left to right)

• Range (bottom to top)

Interval
Notation

closed circles
[ and ]
Open circles
( )
infinity always uses ( )

Domain
(from left to right)

Multiple Choice

A.2(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities

2023 – Q7

• greater than -4, less than or equal to 2

• greater than -2, less than or equal to 3

Multiple Choice

A.2(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities

2022 – Q18

• F

• G

• H

• J

Multiple Choice

A.2(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities

2021 – Q22

• F

• G

• H

• J

​TEKS A.2(B): Write linear equations in two variables in various forms:






Slope-intercept: y = mx + b
Standard: Ax + By = C
Point-slope: y - y₁ = m(x - x₁)
Given: One point and the slope Two points

​Slope-Intercept Form (y = mx + b) Definition: Represents a line with slope m and y-intercept b

Steps:
1) Identify slope (m) and y-intercept (b)
2) Plug into y = mx + b

​Example:
Given: Slope = 2, y-intercept = –3

→ y = 2x – 3

​Given a point and slope:
Point: (4, 5), Slope = –1
→ Use point to solve for b:

5 = –1(4) + b → b = 9 → y = –x + 9

​Standard Form (Ax + By = C)

​Definition: A linear equation where A, B, and C are integers Steps:
1) Start from y = mx + b
2) Rearrange to move all variables to one side

​Example: y = 2x – 3
→ –2x + y = –3
→ Multiply to eliminate fractions or decimals if needed

​From two points: (1, 2) and (3, 6) → Find slope: (6–2)/(3–1) = 2
→ Use y = mx + b
→ y = 2x
→ Rearranged: –2x + y = 0

​Point-Slope Form (y – y₁ = m(x – x₁))
Definition: Used when you know one point and the slope Steps:
1) Identify point (x₁, y₁) and slope m
2) Plug into formula



Example: Point: (–2, 4), Slope = 3
→ y – 4 = 3(x + 2)
→ Optional: Convert to slope-intercept or standard form

​Given Two Points
Steps:
Find slope: m = (y₂ – y₁)/(x₂ – x₁)
Use one point and slope in point-slope form Convert to desired form

Example: Points: (2, 5) and (4, 9)
→ Slope: (9–5)/(4–2) = 2
→ y – 5 = 2(x – 2)
→ Slope-intercept: y = 2x + 1
→ Standard: –2x + y = 1

​Practice Problems

Write the equation of a line through (3, –2) with slope 4 in all three forms.

Write the equation of a line through (–1, 5) and (2, –1).

Convert y = –3x + 7 to standard form.

​STAAR Tips & Common Mistakes
Watch signs when calculating slope
Always simplify fractions
Convert decimals to integers in standard form
Use parentheses carefully in point-slope form

Multiple Choice

A.2(B) write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y₁ = m(x - x₁), given one point and the slope and given two points

2022 – Q25

• A

• B

• C

• D

Multiple Choice

A.2(B) write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y₁ = m(x - x₁), given one point and the slope and given two points

2021 – Q6

• F

• G

• H

• J

​Writing Linear Equations from a Table or Graph

Goal: Write an equation in the form y = mx + b

🔢 From a Table:
1. Pick two points: (x₁, y₁) and (x₂, y₂)
2. Find the slope: m = (y₂ – y₁)/(x₂ – x₁)
3. Use one point to find b
4. Write the equation: y = mx + b

​x	​y
​1	​3
​3	​7

​→ Slope: (7 – 3)/(3 – 1) = 2
→ Use (1, 3): 3 = 2(1) + b
→ b = 1
→ Final: y = 2x + 1

​📊 From a Graph:
Identify two clear points
Find slope and y-intercept
Write: y = mx + b

​Writing from a Verbal Description

Steps:
Identify the slope and a point
Use y = mx + b or point-slope form
Convert to slope-intercept if needed

Example: “A line passes through (–2, 5) with a slope of –3.”
→ Use point-slope: y – 5 = –3(x + 2)
→ Convert: y = –3x – 1

Multiple Choice

A.2(C) write linear equations in two variables given a table of values, a graph, and a verbal description

2023 – Q1

• A

• B

• C

• D

Multiple Choice

A.2(C) write linear equations in two variables given a table of values, a graph, or a verbal description

2023 – Q34

• A

• B

• C

• D