How do you write an equation of the line that passes through (-3,4) and (1,0)?

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How do you write an equation of the line that passes through (-3,4) and (1,0)?

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Answer 1 · 2016-09-02T01:33:41

$y = - x + 1$ $y = -x+1$

Explanation:

The line equation is of the form
$y = a x + b$ $y = ax + b$

As the line passes through this two points
$(x_{0}, y_{0}) = (- 3, 4)$ $(x_0,y_0) = (-3,4)$
$(x_{1}, y_{1}) = (1, 0)$ $(x_1,y_1) = (1,0)$

they both obey the equation
$y_{1} = a x_{1} + b \Rightarrow 0 = a + b \Rightarrow a = - b$ $y_1 = ax_1 + b => 0 = a+b => a = -b$
$y_{0} = a x_{0} + b \Rightarrow 4 = - 3 a + b$ $y_0 = ax_0 + b => 4 = -3a+b$

Therefore:
$4 = - 3 a - a \Rightarrow 4 = - 4 a \Rightarrow a = - 1 and b = 1$ $4 = -3a - a => 4 = -4a => a = -1 and b = 1$

So, the equation of the line that passes through those points is
$y = - x + 1$ $y = -x+1$

Answer 2 · 2016-09-02T01:41:17

We will write the equation in slope-intercept form, $y = m x + c$ $y=mx+c$ . I have calculated below that the slope of the line is -1 and its y-intercept is 1.

The equation of the line can be written: $y = - x + 1$ $y=-x+1$

(we don't write the number '1' in front of the $x$ $x$ , since $1 x = x$ $1x=x$ )

Explanation:

First step: find the slope (gradient) of the line:

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $m=(y_2-y_1)/(x_2-x_1)$

It doesn't matter which of the two given points we decided is 'Point 1' $(x_{1}, y_{1})$ $(x_1,y_1)$ , but let's choose the point $(1, 0)$ $(1,0)$ , so $x_{1} = 1$ $x_1=1$ and $y_{1} = 0$ $y_1=0$ .

Similarly for the other point, $(- 3, 4)$ $(-3,4)$ , so $x_{2} = - 3$ $x_2=-3$ and $y_{2} = 4$ $y_2=4$ .

So $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - 0}{- 3 - 1} = \frac{4}{- 4} = - 1$ $m=(y_2-y_1)/(x_2-x_1)=(4-0)/(-3-1) =4/(-4)=-1$

Now that we know the slope, we can use it and the value of one of the points we were given to find the y-intercept of the line. This is the point where the line crosses the y-axis. The y-axis is the line $x = 0$ $x=0$ , so if we substitute $m = - 1$ $m=-1$ and, for example, $x_{1} = 1$ $x_1=1$ and $y_{1} = 0$ $y_1=0$ , into the equation:

$y = m x + c$ $y=mx+c$

$0 = - 1 (1) + c$ $0=-1(1)+c$

Rearranging, $c = 1$ $c=1$ .

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How do you write an equation of the line that passes through (-3,4) and (1,0)?

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