The standard form of a linear equation, given intercepts, can be derived by first finding the equation using the intercept form and then converting it to standard form. The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A is typically non-negative.
Here's how to obtain the standard form from intercepts:
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Intercept Form: If a line has an x-intercept of 'a' and a y-intercept of 'b', its equation in intercept form is:
x/a + y/b = 1
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Eliminate Fractions: Multiply both sides of the equation by the least common multiple (LCM) of 'a' and 'b' to eliminate the fractions. Let's denote the LCM as 'L'. This gives:
(L/a)x + (L/b)y = L
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Rewrite as Integers: Simplify the coefficients of x and y. Since L is a multiple of both a and b, (L/a) and (L/b) will be integers.
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Rearrange to Standard Form: Ensure the equation is in the form Ax + By = C. If A is negative, multiply the entire equation by -1 to make A positive.
Example:
Suppose a line has an x-intercept of 2 and a y-intercept of 3.
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Intercept Form: x/2 + y/3 = 1
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Eliminate Fractions: The LCM of 2 and 3 is 6. Multiplying both sides by 6:
6(x/2 + y/3) = 6(1)
3x + 2y = 6 -
Standard Form: The equation is already in standard form:
3x + 2y = 6
In this case, A = 3, B = 2, and C = 6.
Key Points:
- The intercept form (x/a + y/b = 1) is a useful starting point when you know the x and y intercepts.
- The standard form (Ax + By = C) requires A, B, and C to be integers, and A is generally non-negative.
- Converting from intercept form to standard form involves eliminating fractions and rearranging the terms.
