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Linear Equations (Unknown on One Side)
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Foundational skills
Identify inverse operations
\[ \text{adding 5} \rightarrow \text{?} \]
Identify inverse operation.
Identify first operation to undo
\[ 3x + 5 = 20 \]
Which to undo first?
One-step equations (addition and subtraction)
Solve one-step addition equations
\[ x + 7 = 12 \]
Solve x + a = b
Solve one-step subtraction equations
\[ x – 4 = 9 \]
Solve x – a = b
One-step with negative answer
\[ x + 8 = 3 \]
Answer is negative integer.
One-step equations (multiplication and division)
Solve one-step multiplication equations
\[ 4x = 20 \]
Solve ax = b
Solve one-step division equations
\[ \frac{x}{3} = 7 \]
Solve x/a = b
One-step mult/div with negative answer
\[ 5x = -15 \]
Answer is negative integer.
One-step with fractional answer
\[ 4x = 7 \]
Answer is a fraction.
Two-step equations (ax + b = c form)
Solve ax + b = c
\[ 3x + 5 = 17 \]
Two-step: multiply then add.
Solve ax – b = c
\[ 2x – 3 = 11 \]
Two-step: multiply then subtract.
Two-step (ax ± b = c) negative answer
\[ 4x + 12 = 0 \]
Answer is negative integer.
Two-step (ax ± b = c) fractional answer
\[ 3x + 2 = 10 \]
Answer is a fraction.
Two-step equations (x/a + b = c form)
Solve x/a + b = c
\[ \frac{x}{4} + 3 = 7 \]
Two-step: divide then add.
Solve x/a – b = c
\[ \frac{x}{3} – 2 = 5 \]
Two-step: divide then subtract.
Two-step (x/a ± b = c) negative answer
\[ \frac{x}{2} + 5 = 2 \]
Answer is negative integer.
Two-step (x/a ± b = c) fractional answer
\[ \frac{x}{3} + 1 = 2\frac{1}{2} \]
Answer is a fraction.
Equations with leading subtraction
Solve a – x = b
\[ 15 – x = 8 \]
Unknown subtracted from constant.
Solve a – bx = c
\[ 20 – 3x = 8 \]
Constant minus multiple of unknown.
Solve a – x/b = c
\[ 10 – \frac{x}{2} = 6 \]
Constant minus unknown divided.
Leading subtraction with negative answer
\[ 5 – 2x = 11 \]
Answer is negative integer.
Leading subtraction with fractional answer
\[ 12 – 5x = 3 \]
Answer is a fraction.
Equations with fraction bar
Solve ax/b = c
\[ \frac{2x}{3} = 4 \]
Multiple of unknown over constant.
Solve (x + a)/b = c
\[ \frac{x + 5}{2} = 7 \]
Sum divided by constant.
Solve (x – a)/b = c
\[ \frac{x – 3}{4} = 2 \]
Difference divided by constant.
Solve (ax + b)/c = d
\[ \frac{2x + 4}{3} = 6 \]
Expression divided by constant.
Fraction bar with negative answer
\[ \frac{x + 8}{2} = 3 \]
Answer is negative integer.
Fraction bar with fractional answer
\[ \frac{x – 2}{3} = 4 \]
Answer is a fraction.
Equations requiring simplification
Simplify before solving (adjacent)
\[ 2x + 3x + 4 = 19 \]
Combine adjacent like terms.
Simplify before solving (non-adjacent)
\[ 3x + 5 + 2x = 20 \]
Combine non-adjacent like terms.
Simplification with negative answer
\[ 2x + x + 9 = 3 \]
Answer is negative integer.
Simplification with fractional answer
\[ 3x + 2x + 3 = 10 \]
Answer is a fraction.
Three-step equations
Solve ax/b + c = d
\[ \frac{2x}{3} + 4 = 10 \]
Three operations to undo.
Solve a – bx/c = d
\[ 12 – \frac{3x}{2} = 6 \]
Leading subtraction with fraction.
Three-step with negative answer
\[ \frac{2x}{3} + 8 = 4 \]
Answer is negative integer.
Three-step with fractional answer
\[ \frac{3x}{2} + 1 = 5 \]
Answer is a fraction.
Special cases
Equation with solution x = 0
\[ 3x + 7 = 7 \]
Solution is zero.
Equation with solution x = 1
\[ 5x + 3 = 8 \]
Solution is one.
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