Packets  /  Linear Equations
Grades 7-9Algebra

Linear Equations

Solve one-step, two-step, and multi-step linear equations, handle variables on both sides, then master slope, slope-intercept form, writing equations, and real-world word problems.

1Solving One-Step Equations

Key idea: Undo what is done to x: addition undoes subtraction, multiplication undoes division — and always do the same thing to BOTH sides.
  1. Identify the operation acting on x.
  2. Apply the inverse (opposite) operation to both sides of the equation.
  3. Simplify to isolate x.
  4. Check your answer in the original equation.
Worked example
x + 7 = 15
7 is added to x, so subtract 7 from both sides: x = 15 – 7 = 8. Check: 8 + 7 = 15.
Worked example
4x = 36
x is multiplied by 4, so divide both sides by 4: x = 36 / 4 = 9. Check: 4(9) = 36.
Watch out: Dividing both sides by a number means dividing EVERY term on that side — you cannot divide just one term.

2Solving Two-Step Equations

Key idea: Two-step equations hide x behind two operations — undo them in reverse order: addition/subtraction first, then multiplication/division.
  1. Undo the added or subtracted number first (both sides).
  2. Undo the multiplied or divided number second.
  3. Simplify and isolate x.
  4. Check by substituting back.
Worked example
2x + 5 = 13
Undo the +5 first: 2x = 13 – 5 = 8. Then undo the times-2: x = 8 / 2 = 4. Check: 2(4) + 5 = 13.
Worked example
(x/4) – 3 = 2
Undo the -3 first: x/4 = 2 + 3 = 5. Then undo the divide-by-4: x = 5(4) = 20. Check: 20/4 – 3 = 5 – 3 = 2.
Watch out: Do NOT divide first in 2x + 5 = 13 — dividing both sides by 2 gives x + 5/2 = 13/2, which is legal but much messier. Addition/subtraction first is the clean path.

3Multi-Step Equations

Key idea: Clean up the equation first — distribute and combine like terms — until it becomes a two-step equation you already know how to solve.
  1. Distribute to remove parentheses.
  2. Combine like terms on each side.
  3. Now you have a two-step equation: solve it.
  4. Check your answer in the original equation.
Worked example
3(x + 4) = 30
Distribute: 3x + 12 = 30. Now a two-step equation: 3x = 18, x = 6. Check: 3(6 + 4) = 3(10) = 30.
Worked example
7x + 3x – 5 = 25
Combine like terms: 10x – 5 = 25. Then 10x = 30, x = 3. Check: 7(3) + 3(3) – 5 = 21 + 9 – 5 = 25.
Watch out: When you distribute, multiply the number by EVERY term inside the parentheses: 3(x + 4) = 3x + 12, NOT 3x + 4.

4Variables on Both Sides

Key idea: Get all the x's on one side and all the plain numbers on the other, then solve the two-step equation that is left.
  1. Distribute and combine like terms on each side if needed.
  2. Add or subtract to move every x-term to one side.
  3. Add or subtract to move every constant to the other side.
  4. Divide by the coefficient of x and check.
Worked example
6x – 4 = 3x + 11
Move variables left: subtract 3x from both sides, 3x – 4 = 11. Move constants right: add 4, 3x = 15. So x = 5. Check: 6(5) – 4 = 26 and 3(5) + 11 = 26.
Worked example
2(x + 3) = x + 9
Distribute: 2x + 6 = x + 9. Subtract x: x + 6 = 9. Subtract 6: x = 3. Check: 2(3 + 3) = 12 and 3 + 9 = 12.
Watch out: If the x's cancel and you get a true statement like 5 = 5, there are infinitely many solutions. If you get a false statement like 3 = 7, there is NO solution.

5Slope from Two Points

Key idea: Slope m = (y2 – y1) / (x2 – x1) — rise over run. The same two points always give the same slope no matter which point you call point 1.
  1. Label the points (x1, y1) and (x2, y2).
  2. Compute rise = y2 – y1 (change in y).
  3. Compute run = x2 – x1 (change in x).
  4. Slope m = rise / run. Simplify the fraction.
Worked example
Slope through (2, 3) and (6, 11)
m = (11 – 3) / (6 – 2) = 8 / 4 = 2. The line rises 2 for every 1 it runs.
Worked example
Slope through (1, 4) and (5, -8)
m = (-8 – 4) / (5 – 1) = -12 / 4 = -3. Negative slope means the line falls as it moves right.
Watch out: Keep the point order the same on top and bottom: (y2 – y1)/(x2 – x1). Flipping only the top gives the wrong sign.

6Slope from Graphs: Horizontal vs Vertical

Key idea: Zero over a number is 0 (horizontal line). A number over zero is UNDEFINED (vertical line). They are opposite extremes — do not mix them up!
  1. Compute rise and run as usual.
  2. If the rise is 0, the slope is 0 and the line is horizontal: y = a number.
  3. If the run is 0, the slope is undefined and the line is vertical: x = a number.
  4. On a graph, count rise/run between two grid points and simplify.
Worked example
Line through (-2, 5) and (4, 5)
m = (5 – 5) / (4 – (-2)) = 0 / 6 = 0. Zero slope means the line is HORIZONTAL.
Worked example
Line through (3, -1) and (3, 4)
m = (4 – (-1)) / (3 – 3) = 5 / 0, which is UNDEFINED. Number over zero means the line is VERTICAL.
Watch out: 'Undefined slope' does NOT mean slope 0. Horizontal: y = 5 (slope 0). Vertical: x = 3 (no slope). They look and behave completely differently.

7Slope-Intercept Form

Key idea: In y = mx + b, m is the slope and b is where the line crosses the y-axis. To find the slope from standard form Ax + By = C, solve for y — the slope is -A/B.
  1. Check the equation is in the form y = mx + b.
  2. Read off m (slope) as the coefficient of x.
  3. Read off b (y-intercept): the line crosses the y-axis at (0, b).
  4. For Ax + By = C, solve for y first, then read m = -A/B.
Worked example
y = 4x – 7
In y = mx + b, m = 4 is the slope and b = -7 is the y-intercept. The line crosses the y-axis at (0, -7).
Worked example
3x + 2y = 12
Solve for y: 2y = -3x + 12, so y = -32x + 6. The slope is the coefficient of x: m = -3/2, and the y-intercept is 6.
Watch out: In 3x + 2y = 12 the slope is NOT 3. Solve for y first! The slope is -3/2 — the negative sign comes from moving 3x across.

8Writing Equations of Lines

Key idea: Find the slope first, then use point-slope form y – y1 = m(x – x1) and simplify to y = mx + b. Parallel lines share the slope; perpendicular slopes are negative reciprocals (they multiply to -1).
  1. Compute or identify the slope m (for perpendicular, take the negative reciprocal).
  2. Plug m and the point (x1, y1) into y – y1 = m(x – x1).
  3. Distribute and solve for y.
  4. Check: the point must satisfy your equation.
Worked example
Line through (4, 5) with slope 2
Point-slope: y – 5 = 2(x – 4). Distribute: y – 5 = 2x – 8. Add 5: y = 2x – 3.
Worked example
Line perpendicular to y = 2x + 5 through (4, 1)
Perpendicular slope is the negative reciprocal of 2: m = -1/2. Point-slope: y – 1 = -12(x – 4) = -12x + 2. So y = -12x + 3.
Watch out: Parallel lines have the SAME slope; perpendicular slopes FLIP and change sign (2 becomes -1/2, not just -2). Forgetting the flip is the classic error.

9Word Problems

Key idea: Name the unknown, turn the words into an equation, solve it, and translate the answer back into the story's units.
  1. Define a variable for the unknown quantity.
  2. Translate each part of the sentence into math.
  3. Solve the equation.
  4. Check that the answer makes sense in the story.
Worked example
A taxi charges $3 plus $2 per mile. A ride costs $21. How many miles?
Let m = miles. 3 + 2m = 21. Subtract 3: 2m = 18. Divide: m = 9. The ride was 9 miles. Check: 3 + 2(9) = 21.
Worked example
A phone plan costs $25 per month plus $0.10 per minute. The bill is $40. How many minutes were used?
Let t = minutes. 25 + 0.10t = 40. Subtract 25: 0.10t = 15. Divide: t = 150 minutes. Check: 25 + 0.10(150) = 25 + 15 = 40.
Watch out: Always finish with a sentence that answers the actual question ('9 miles'), not just 'x = 9' — and make sure the answer is realistic (a negative number of miles means something went wrong).
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60-Second Challenge

How many linear equations problems can you solve in 60 seconds?