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Implicit Form (General Form ax + by = c)
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Foundational Skills
Identify coefficients in ax + by = c
\[ 2x + 3y = 12 \rightarrow a, b, c \]
Identify the values of a, b, and c from an equation in general form.
Identify coefficients when signs vary
\[ 3x – 2y = 6 \rightarrow a, b, c \]
Identify coefficients when the equation has mixed signs.
Recognise equivalent forms
\[ 2x + 4y = 8 \text{ vs } x + 2y = 4 \]
Recognise when two general form equations represent the same line.
Write with positive a
\[ -2x + 3y = 6 \rightarrow 2x – 3y = -6 \]
Rewrite an equation so the x-coefficient is positive.
Gradient is not coefficient of x
\[ 2x + y = 5: m \neq 2 \]
Recognise that the gradient in general form is -a/b not a.
Calculate gradient from general form
\[ 3x + 2y = 10 \rightarrow m = -\frac{3}{2} \]
Calculate gradient by rearranging or using -a/b.
Compare gradients between forms
\[ y = 2x + 3 \text{ match to } ax + by = c \]
Match a y = mx + c equation to a general form equation with same gradient.
Finding Intercepts
Find y-intercept
\[ 2x + 3y = 12 \rightarrow y\text{-int} = 4 \]
Find where line crosses y-axis by setting x = 0.
Find x-intercept
\[ 2x + 3y = 12 \rightarrow x\text{-int} = 6 \]
Find where line crosses x-axis by setting y = 0.
Find both intercepts
\[ 4x + 2y = 8 \rightarrow (2,0), (0,4) \]
Find both x and y intercepts from a general form equation.
Find fractional intercept
\[ 3x + 2y = 5 \rightarrow y\text{-int} = \frac{5}{2} \]
Find intercept when result is a fraction.
Identify intercepts as unknowns
\[ 4x + 5y = 20: (p,0), (0,q) \]
Find both intercepts expressed as unknowns.
Converting Forms
y = mx + c to general (positive m)
\[ y = 2x + 3 \rightarrow ax + by = c \]
Convert from slope-intercept form to general form.
y = mx + c to general (negative m)
\[ y = -3x + 5 \rightarrow ax + by = c \]
Convert when gradient is negative.
y = mx + c to general (fractional m)
\[ y = \frac{1}{2}x + 3 \rightarrow ax + by = c \]
Convert from fractional gradient to integer coefficient form.
y = mx + c to general (neg. frac. m)
\[ y = -\frac{2}{3}x + 4 \rightarrow ax + by = c \]
Convert when gradient is negative fraction.
General to y = mx + c (integer)
\[ 2x + y = 7 \rightarrow y = -2x + 7 \]
Convert from general form to slope-intercept form.
General to y = mx + c (fractional)
\[ 2x + 3y = 9 \rightarrow y = mx + c \]
Convert when result has fractional gradient.
General to y = mx + c (negatives)
\[ 3x – 2y = 8 \rightarrow y = mx + c \]
Convert when original has negative coefficient.
Simplify to lowest terms
\[ 4x + 6y = 10 \rightarrow 2x + 3y = 5 \]
Simplify a general form equation by dividing by HCF.
Verify conversion is correct
\[ y = 3x – 2 \rightarrow 3x – y = 2 \text{ ?} \]
Check if a conversion between forms is correct.
Writing Equations in General Form
From intercepts
\[ (4,0), (0,3) \rightarrow ax + by = c \]
Write general form equation from both intercepts.
From gradient and y-intercept
\[ m = -\frac{2}{3}, c = 4 \rightarrow ax + by = c \]
Write general form from gradient and y-intercept.
From gradient and point
\[ m = 2, (3,1) \rightarrow ax + by = c \]
Write general form from gradient and a point.
From two points (integer m)
\[ (1,2), (3,8) \rightarrow ax + by = c \]
Write general form from two points.
From two points (fractional m)
\[ (0,1), (4,3) \rightarrow ax + by = c \]
Write general form from two points when gradient is fractional.
Sketching and Interpreting
Identify equation from graph
\[ \text{Graph} \rightarrow ax + by = c \]
Read intercepts from graph and write general form equation.
Point on line?
\[ (3,2) \text{ on } 2x + 3y = 12 \text{ ?} \]
Check if a point satisfies a general form equation.
Applications
Interpret general form in context
\[ 50t + d = 200: d \text{ when } t = 0 \]
Find initial value from general form equation in context.
Form equation from intercepts
\[ \text{Tank: } 120L, \text{ empty in } 30min \]
Form general form equation from real-world intercepts.
Compare lines in different forms
\[ y = 2x + 3 \text{ vs } 4x – 2y = -6 \]
Compare lines given in different forms by converting.
Extensions
Specified coefficient
\[ (2,5), m=3 \rightarrow ax + by = c, a=3 \]
Write general form with a constraint on one coefficient.
Find intersection of two lines
\[ 2x + y = 7, x + 3y = 11 \]
Solve simultaneous equations both in general form.
Find perpendicular line
\[ \perp \text{ to } 2x + 3y = 6, (0,0) \]
Write perpendicular line equation in general form.
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