TEKSForge
Lesson library
A.2B·Algebra 1·Linear FunctionsReadiness

Writing Linear Equations

Write linear equations in slope-intercept form given a slope and a point, or two points.

Students write linear equations from a slope and point or from two points, using substitution to determine the intercept and verifying both given points.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Practice this skill Ask the tutorMy Algebra 1 Study Plan Real-world applications45 minutes

Learn the concept

Find the rate

From two points, calculate m=(y2−y1)/(x2−x1)m=(y_2-y_1)/(x_2-x_1) with the same subtraction order in numerator and denominator. If x-values are equal, the line is vertical and cannot be written as y=mx+b.

Use a point

Given slope and a point, substitute the point's coordinates into y=mx+by=mx+b and solve for b. Alternatively use y−y1=m(x−x1)y-y_1=m(x-x_1), then simplify. The given point's y-coordinate equals b only when its x-coordinate is zero.

Verify the equation

Substitute both given points into the final equation. A correct line must contain both, not just have the correct slope. In contexts, attach units to the rate and initial value.

Visualize the concept

Original TEKSForge instructional diagrams, designed for this TEKS skill and clearly distinguished from released STAAR artwork.

A line passes through negative 1,negative 3 and 2,3 with y-intercept 0,negative 1; a slope triangle marks run 3 and rise 6.
Writing a linear equation from two pointsOriginal TEKSForge instructional visual · not a released STAAR figure

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Released STAAR Algebra I EOC · 2022 · Item 25 · 1 point

A coordinate grid with points plotted at (-2, 6) and (1, -4).
Original visual from the released STAAR test.

Worked example

Write an equation through (2,5) and (6,13).

  1. Find slope: (13−5)/(6−2)=8/4=2(13-5)/(6-2)=8/4=2.
  2. Use (2,5): 5=2(2)+b5=2(2)+b, so b=1.
  3. Write y=2x+1y=2x+1 and check the other point: 13=2(6)+113=2(6)+1.

y=2x+1y=2x+1

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 1 Writing Linear Equations Given a Point and the Slope

Five Minute Math · Aaron Daffern, Dallas ISD

Use a point and slope to write a linear equation in point-slope or slope-intercept form.

Watch on YouTube
Texas public school video

Algebra 1 Writing Linear Equations Given Two Points

Five Minute Math · Aaron Daffern, Dallas ISD

Calculate slope from two points, write the equation, and verify both points.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. Write a line with slope −3 through (2,1).

    Show answer and reasoning

    y=−3x+7y=-3x+7

    Substitute: 1=−3(2)+b, so b=7. The intercept is not the point's y-value of 1.

  2. 2. Write a line through (1,4) and (3,10).

    Show answer and reasoning

    y=3x+1y=3x+1

    Slope is (10−4)/(3−1)=3. Substituting (1,4) yields b=1; both points satisfy the equation.

Released STAAR practice

Texas Education Agency · released EOC items · explanations adapted from TEA rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Released STAAR Algebra I EOC · 2019 · Item 6 · 1 point

What is the equation in slope-intercept form of the line that passes through the points (−4,2)(-4, 2) and (12,6)(12, 6)?

  1. y=0.25x+3y = 0.25x + 3
  2. y=0.25x−4.5y = 0.25x - 4.5
  3. y=4x+18y = 4x + 18
  4. y=4x−42y = 4x - 42
Show STAAR answer and explanation

y=0.25x+3y = 0.25x + 3

Slope: m=6−212−(−4)=416=0.25m = \frac{6 - 2}{12 - (-4)} = \frac{4}{16} = 0.25. Use (12,6)(12, 6): y−6=0.25(x−12)y - 6 = 0.25(x - 12), so y=0.25x+3y = 0.25x + 3.

y=0.25x−4.5y = 0.25x - 4.5: The slope is right, but the x- and y-coordinates of (−4,2)(-4, 2) were swapped when finding b.

y=4x+18y = 4x + 18: The slope was flipped: run over rise instead of rise over run. Use y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}.

y=4x−42y = 4x - 42: The slope was flipped. Use change in y over change in x.

Ask the tutor about this question

Released STAAR Algebra I EOC · 2022 · Item 25 · 1 point

A coordinate grid with points plotted at (-2, 6) and (1, -4).
Original visual from the released STAAR test.

Two points are plotted on the grid. Which equation in slope-intercept form best represents the line that passes through these two points?

  1. y=−23x−113y = -\frac{2}{3}x - \frac{11}{3}
  2. y=−113x−23y = -\frac{11}{3}x - \frac{2}{3}
  3. y=−23x−103y = -\frac{2}{3}x - \frac{10}{3}
  4. y=−103x−23y = -\frac{10}{3}x - \frac{2}{3}
Show STAAR answer and explanation

y=−103x−23y = -\frac{10}{3}x - \frac{2}{3}

Slope: m=−4−61−(−2)=−103m = \frac{-4 - 6}{1 - (-2)} = -\frac{10}{3}. Use (−2,6)(-2, 6): 6=−103(−2)+b6 = -\frac{10}{3}(-2) + b, so b=6−203=−23b = 6 - \frac{20}{3} = -\frac{2}{3}.

y=−23x−113y = -\frac{2}{3}x - \frac{11}{3}: The slope and y-intercept are mixed up. Find the slope first with rise over run.

y=−113x−23y = -\frac{11}{3}x - \frac{2}{3}: The y-intercept is right, but the slope is not. Recheck −4−61−(−2)\frac{-4 - 6}{1 - (-2)}.

y=−23x−103y = -\frac{2}{3}x - \frac{10}{3}: The slope and y-intercept were switched. The slope is −103-\frac{10}{3}.

Ask the tutor about this question

Released STAAR Algebra I EOC · 2021 · Item 6 · 1 point

What is the equation in slope-intercept form of the line that passes through the points (−26,−11)(-26,-11) and (39,34)(39,34)?

  1. y=−913x+7y = -\frac{9}{13}x + 7
  2. y=−913x−7y = -\frac{9}{13}x - 7
  3. y=913x+7y = \frac{9}{13}x + 7
  4. y=913x−7y = \frac{9}{13}x - 7
Show STAAR answer and explanation

y=913x+7y = \frac{9}{13}x + 7

Slope: 34−(−11)39−(−26)=4565=913\frac{34-(-11)}{39-(-26)}=\frac{45}{65}=\frac{9}{13}. Using (39,34)(39,34) gives 34=913(39)+b=27+b34=\frac{9}{13}(39)+b=27+b, so b=7b=7.

y=−913x+7y = -\frac{9}{13}x + 7: The slope is positive because both x and y increase.

y=−913x−7y = -\frac{9}{13}x - 7: Both the slope and y-intercept signs are incorrect.

y=913x−7y = \frac{9}{13}x - 7: The slope is correct, but substituting (39,34)(39,34) gives b=7b=7.

Ask the tutor about this question

Released STAAR Algebra I EOC · 2025 · Item 8 · 1 point

What is the equation in slope-intercept form of the line that has a slope of −211-\frac{2}{11} and passes through the point (22,5)(22, 5)?

  1. y=−211x+9y = -\frac{2}{11}x + 9
  2. y=−211x+1y = -\frac{2}{11}x + 1
  3. y=−211x−9y = -\frac{2}{11}x - 9
  4. y=−211x−1y = -\frac{2}{11}x - 1
Show STAAR answer and explanation

y=−211x+9y = -\frac{2}{11}x + 9

Substitute the point (22,5)(22, 5) and m=−211m = -\frac{2}{11} into y=mx+by = mx + b: 5=−211(22)+b5 = -\frac{2}{11}(22) + b, so 5=−4+b5 = -4 + b and b=9b = 9. The equation is y=−211x+9y = -\frac{2}{11}x + 9.

y=−211x+1y = -\frac{2}{11}x + 1: 4 was subtracted from the left side instead of added when solving 5=−4+b5 = -4 + b, giving b=1b = 1.

y=−211x−9y = -\frac{2}{11}x - 9: A sign error when solving for bb gave b=−9b = -9.

y=−211x−1y = -\frac{2}{11}x - 1: 5 and 4 were subtracted instead of added when solving for bb, giving b=−1b = -1.

Step-by-step walkthrough Ask the tutor about this question
Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Calculate slope from two coordinate pairs.
  • Use a known slope and point to solve for b in y=mx+by=mx+b.
  • Verify that a completed equation contains every given point.

Instructional sequence

  1. 1

    Engage

    5 min

    Ask why knowing only a slope does not identify one unique line.

  2. 2

    Direct instruction

    10 min

    Model m=(y2−y1)/(x2−x1)m=(y_2-y_1)/(x_2-x_1), then substitute x, y, and m to solve for b.

  3. 3

    Guided practice

    12 min

    Write an equation from slope 3 and (2,7)(2,7), then from (1,2)(1,2) and (3,8)(3,8), narrating what each substitution accomplishes.

  4. 4

    Independent application

    12 min

    Students write equations and verify each point by substitution, correcting slope or sign errors before moving on.

  5. 5

    Closure

    6 min

    Complete the exit ticket with a full verification line.

Misconceptions to address

“The y-coordinate of the given point is automatically b.”

Teacher move: b is the value when x=0; substitute the point's x and y to calculate it.

“Slope is change in x divided by change in y.”

Teacher move: Keep vertical change over horizontal change and subtract coordinates in a consistent order.