Forms of Linear Equations

The point-slope form is used 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 which passes through the point \( (3, 5) \) and has slope \( - 2 \).

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From the point-slope form, the equation is \( ( y - 5) = -2 ( x - 3) \), which can be simplified as

\[ y = -2x + 11 . \ _\square \]

The point-point form is used when we know \(2\) points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line. Given this information, we know that the slope is equal to \( m = \frac{ y_2 - y_1} { x_2 - x_1} \), and hence the equation using the point-slope form is

\[ ( y - y_1) = \frac{ y_2 - y_1 } { x_2 - x_1} \times ( x - x_1 ). \]

Find the equation of the line which passes through the points \( P = (1, 7) \) and \( Q = ( -1 , -3 ) \).

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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 \( y = 5x +2 \). \(_\square\)

Find the equation of the line which passes through the points \( P = (0, 3) \) and \( Q = ( -10 , 0) \).

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The slope of the line is \( m = \frac{ 0 - 3 } { -10 - 0 } = \frac{-3} {-10} = \frac{3}{10} \). Hence, the equation of the line is \( (y - 3) = \frac{3}{10} ( x - 0) \), or \( y = \frac{3}{10} x +3.\ _\square\)

The slope-intercept form is used when we know the slope \(m\) and the \(y\)-intercept \( (0, b ) \). Given this information, the equation of the line is simply

\[ y = mx + b. \]

This is actually a special case of the point-slope form, where we use the point \( ( x_1, y_1 ) = (0,b) \).

Note that the reason these constants embody the values of the slope and the \(y\)-intercept are evident in the equation itself. In the case of \( m,\) since \(m\) is multiplied by \( x \), any change in \(x\) will result in an \( m \times x \) change in \( y\). But that idea—the relationship between the change in \(x \) and the subsequent change in \( y\)—is the essence of slope. As to why \(b\) is the value of the \(y\)-intercept, the \(y\)-intercept occurs when the graph touches the \(y\)-axis (i.e., when \(x =0\)). But we can see from the equation above that when \(x=0\), \(y=b\).

What is the slope of \( y = 2x +3 \)?

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Solution 1:
Since the slope is the value of \(m\), we can see that the slope of this equation is \(2.\ _\square \)

Solution 2:
Choosing two points on the graph of \( y = 2x+3 \), we have \( (0,3) \) and \( (1,5) \). Evaluating the slope directly from these points, we see that \( m = \frac{5-3}{2-1} = 2 \). Thus the slope is \(2.\ _\square \)

What is the \(y\)-intercept of \( y = -4x +10 \)?

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Since the \(y\)-intercept occurs when \( x = 0 \), we can see that the \(y\)-intercept is \(10.\ _\square \)

The intercept-intercept form is used when we know the \(2\) intercepts of the line: the \(x\)-intercept of \( (a,0) \) and the \(y\)-intercept of \( (0,b) \). Given this information, the equation of the line is

\[\frac{x}{a} +\frac{y}{b} = 1.\]

This is actually a special case of the point-point form, where we use the \(2\) points as above. We would have obtained that the slope of the given line is \(\frac{b-0}{0-a}=\frac{-b}{a},\) and thus the equation is \( (y-0) = \frac{-b}{a} (x - b) \). Then, rearranging the terms would give us the above equation.

If the \(x\)-intercept and \(y\)-intercept of a line are \((5,0)\) and \((0,6)\), respectively, then what is the equation of the line?

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We have \(\frac{x}{5} + \frac{y}{6} = 1.\) To simplify this, we can multiply both sides by \(30\) and get \(6x + 5y = 30\). \(_\square\)