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Linear Equations: Unknown on Both Sides
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Foundational Skills
Identify which side has more x terms
\[ 5x + 3 = 2x + 9 \]
Identify which side has the larger coefficient of x (positive coefficients).
Identify which side has more x terms (negative coefficients)
\[ -2x + 5 = -5x + 8 \]
Identify which side has more x’s when coefficients can be negative.
Identify the first step when x is on both sides
\[ 4x + 2 = x + 8 \]
Recognise that the first step is to collect x terms onto one side.
Basic Equations (ax + b = cx + d)
Solve ax + b = cx + d (larger coefficient on left)
\[ 5x + 3 = 2x + 15 \]
Solve equations where the left side has the larger x coefficient.
Solve ax + b = cx + d (larger coefficient on right)
\[ 2x + 12 = 6x + 4 \]
Solve equations where the right side has the larger x coefficient.
Solve ax – b = cx + d
\[ 6x – 4 = 3x + 8 \]
Solve equations with subtraction on the left and addition on the right.
Solve ax + b = cx – d
\[ 4x + 10 = 7x – 5 \]
Solve equations with addition on the left and subtraction on the right.
Solve ax – b = cx – d
\[ 5x – 3 = 2x – 9 \]
Solve equations with subtraction on both sides.
Basic equation with negative answer
\[ 3x + 10 = x + 2 \]
Solve equations where the answer is a negative integer.
Basic equation with fractional answer
\[ 5x + 2 = 2x + 4 \]
Solve equations where the answer is a fraction.
Single x Term on One Side
Solve ax + b = x + d
\[ 5x + 3 = x + 15 \]
Solve equations with coefficient 1 on the right.
Solve ax + b = d – x
\[ 3x + 2 = 14 – x \]
Solve equations with negative coefficient on one side.
Solve ax – b = d – x
\[ 4x – 3 = 17 – x \]
Solve equations with subtraction on both sides.
Single x term with negative answer
\[ 5x + 8 = x – 4 \]
Solve equations where the answer is negative.
Equations with Leading Constant
Solve b + ax = cx + d
\[ 5 + 4x = 2x + 11 \]
Solve equations with constant before x on the left.
Solve b + ax = d + cx
\[ 3 + 5x = 12 + 2x \]
Solve equations with constants before x on both sides.
Solve a = bx + c – dx
\[ 20 = 5x + 8 – 2x \]
Solve equations where one side has two x terms.
Leading constant with fractional answer
\[ 4 + 5x = 2x + 6 \]
Solve leading constant equations where the answer is a fraction.
Equations Requiring Simplification
Simplify left side then solve
\[ 2x + 3x + 4 = 4x + 9 \]
Solve equations requiring like terms to be collected on the left.
Simplify right side then solve
\[ 6x + 2 = 2x + x + 11 \]
Solve equations requiring like terms to be collected on the right.
Simplify both sides then solve
\[ 2x + 3x + 4 = x + 2x + 10 \]
Solve equations requiring like terms to be collected on both sides.
Simplification equations with negative answer
\[ 3x + 2x + 15 = 2x + 6 \]
Solve equations requiring simplification where the answer is negative.
Equations with Negative Coefficients
Solve ax + b = -cx + d
\[ 3x + 4 = -2x + 19 \]
Solve equations where one x coefficient is explicitly negative.
Solve -ax + b = cx + d
\[ -2x + 15 = 3x + 5 \]
Solve equations where the left x coefficient is explicitly negative.
Solve -ax + b = -cx + d
\[ -2x + 10 = -5x + 19 \]
Solve equations where both x coefficients are explicitly negative.
Solve ax + b = d – cx
\[ 3x + 5 = 20 – 2x \]
Solve equations with constant minus x-term on the right.
Solve b – ax = cx + d
\[ 18 – 2x = 3x + 3 \]
Solve equations with constant minus x-term on the left.
Solve b – ax = d – cx
\[ 15 – 2x = 9 – 5x \]
Solve equations with constant minus x-term on both sides.
Negative coefficient with negative answer
\[ 2x + 3 = -3x – 12 \]
Solve negative coefficient equations where the answer is negative.
Negative coefficient with fractional answer
\[ 3x + 1 = -2x + 4 \]
Solve negative coefficient equations where the answer is a fraction.
Larger Coefficients
Solve equations with larger coefficients
\[ 9x + 15 = 5x + 43 \]
Solve equations with x on both sides using larger coefficients.
Larger coefficient equations with negative answer
\[ 8x + 30 = 5x + 9 \]
Solve equations with larger coefficients where the answer is negative.
Special Cases
Equation with solution x = 0
\[ 5x + 7 = 3x + 7 \]
Solve equations where the solution is zero.
Equation with solution x = 1
\[ 6x + 4 = 2x + 8 \]
Solve equations where the solution is one.
Equation with equal x coefficients (no solution)
\[ 4x + 5 = 4x + 9 \]
Recognise when an equation has no solution.
Equation with equal x coefficients (infinite solutions)
\[ 3x + 7 = 3x + 7 \]
Recognise when an equation has infinitely many solutions.
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