Forms of Linear Equations
The three major forms of linear equations are slope-intercept form, point-slope form, and standard form.
Slope Intercept Form
The slope-intercept form is \[ y = mx + b. \]
It provides us with two important pieces of information about the graph of a line: the slope \(m\) and the \(y\)-intercept \( (0, b ) \).
What is the slope of \( y = 2x +3 \)?
Since the slope is the value of \(m\), we can see that the slope of this equation is \(2.\ _\square \)
What is the \(y\)-intercept of \( y = -4x +10 \)?
Since the \(y\)-intercept occurs when \( x = 0 \), we can see that the \(y\)-intercept is \(10.\ _\square \)
Point-Slope Form
We use point-slope form when we know a point \( (x_1, y_1) \) on the line, and the slope \(m\). Given this information, the equation of the line is
\[ ( y - y_1) = m ( x - x_1). \]
Find the equation of the straight line that passes through the point \( (3, 5) \) and has slope \( - 2 \).
From the point-slope form, the equation is \( ( y - 5) = -2 ( x - 3) \), which can be simplified as\[ y = -2x + 11 . \ _\square \]
Find the equation of the line which passes through the points \( P = (1, 7) \) and \( Q = ( -1 , -3 ) \).
The slope of the line is \( m = \frac{ 7 - ( -3) } { 1 - (-1) } = \frac{ 10} { 2} = 5 \). Hence, the equation of the line is \( (y - 7) = 5 ( x - 1) \), or \( y = 5x +2 \). \(_\square\)
Standard Form
The standard form of a line is \(Ax+By=C.\) (A, B,) and \(C\) are integers.
This form of a line is particularly useful for determining both the \(x\)- and \(y\)-intercepts. We can determine the \(x\)-intercept of the line by substituting 0 for \(y\) and solving for \(x.\) We can determine the \(y\)-intercept of the line by substituting 0 for \(x\) and solving for \(y.\)
If the equation of a line is \(3x + 5y = 60,\) what are the \(x\)-intercept and \(y\)-intercept of the line?
To find the \(x\)-intercept, we substitute 0 for \(y\) and solve: \[\begin{align} 3x + 5(0) &= 60 \\ 3x &= 60 \\ x &= 20.\end{align}\]
To find the \(y\)-intercept, we substitute 0 for \(x\) and solve: \[\begin{align} 3(0) + 5y &= 60 \\ 5y &= 60 \\ x &= 12.\end{align}\]
The \(x\)-intercept is \((12,0)\) and the \(y\)-intercept is \((0,20).\)
If the \(x\)-intercept and \(y\)-intercept of a line are \((5,0)\) and \((0,6)\), respectively, what is the equation of the line?
Dividing both sides of the standard form equation by \(C\) yields the equation \(\frac{A}{C}x+\frac{B}{C}y=1.\) Given this equation, the \(x\)-intercept is \(\left(\frac{C}{A},0\right)\) and the \(y\)-intercept is \(\left(0,\frac{B}{C}\right).\)
Since our \(x\)-intercept is 5, \(\frac{A}{C} = \frac{1}{5}.\) Since our \(y\)-intercept is 6, \(\frac{B}{C} = \frac{1}{6}.\)
Substituting our known values into the equation, we have \(\frac{1}{5}x + \frac{1}{6}y = 1.\) Multiplying both sides by \(30\) yields \(6x + 5y = 30\). \(_\square\)