Algebra 1 Practice - Write an Equation in Standard Form (Example 1)
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To write an equation in standard form \(Ax + By = C\) given information about points and slopes, follow these steps:
1. *Using a Point and the Slope*
If you have the slope \(m\) and a point \((x_1, y_1)\), you can convert this information into standard form:
1. *Start with the Slope-Intercept Form:*
Convert the given slope \(m\) and point \((x_1, y_1)\) to slope-intercept form \(y = mx + b\) to find the y-intercept \(b\).
\[ b = y_1 - mx_1 \]
Substitute \(m\) and \((x_1, y_1)\) into:
\[ y = mx + b \]
2. *Convert to Standard Form:*
Rearrange \(y = mx + b\) into standard form \(Ax + By = C\).
Start by moving the \(mx\) term to the left side:
\[ -mx + y = b \]
Multiply through by -1 to make \(A\) positive if needed:
\[ mx - y = -b \]
Rewrite in the form \(Ax + By = C\):
\[ Ax + By = C \]
*Example:*
Slope (\(m\)) = 2
Point = (3, 4)
Find \(b\):
\[ b = 4 - 2 \cdot 3 \]
\[ b = 4 - 6 \]
\[ b = -2 \]
The equation in slope-intercept form is:
\[ y = 2x - 2 \]
Convert to standard form:
\[ 2x - y = 2 \]
2. *Using Two Points*
If you have two points \((x_1, y_1)\) and \((x_2, y_2)\):
1. *Calculate the Slope \(m\):*
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
2. *Find the Equation in Point-Slope Form:*
Use one of the points \((x_1, y_1)\) and the calculated slope \(m\) in the point-slope form:
\[ y - y_1 = m(x - x_1) \]
3. *Convert to Standard Form:*
Rearrange the point-slope form into standard form \(Ax + By = C\):
Distribute \(m\) and simplify:
\[ y - y_1 = m(x - x_1) \]
\[ y - y_1 = mx - mx_1 \]
\[ -mx + y = -mx_1 + y_1 \]
Rewrite in standard form:
\[ mx - y = mx_1 - y_1 \]
*Example:*
Points = (2, 1) and (4, 5)
Calculate the slope \(m\):
\[ m = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2 \]
Use point (2, 1):
\[ y - 1 = 2(x - 2) \]
\[ y - 1 = 2x - 4 \]
\[ -2x + y = -4 + 1 \]
\[ -2x + y = -3 \]
To make \(A\) positive, multiply through by -1:
\[ 2x - y = 3 \]
Using these methods, you can convert given points and slopes into standard form equations \(Ax + By = C\).
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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa
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