Finding the Equation of a Straight Line
🎬 Video: How to Find y = mx + c
How to Find the Slope (m) of a Straight Line? (0:01)
To find the equation of a straight line, we use the formula:
$$ y = mx + c $$
We start by finding the slope ($m$), using the formula:
$$ m = \frac{y_2\, -\, y_1}{x_2 \,-\, x_1} $$
We are given two points on the line:
$$
\begin{aligned}
P_1 &= (2, 3) \\
P_2 &= (4, 7)
\end{aligned}
$$
Substitute the values into the formula:
$$ m = \frac{7\, – \,3}{4\, – \,2} = \frac{4}{2} = 2 $$
So, the slope is $m = 2$.
How to Find the Y-intercept (c) of a Straight Line? (1:06)
Now that we’ve found the slope $m = 2$, we plug it into the equation $y = mx + c$.
This gives us:
$$y=2x+c$$
To find the value of $c$, choose one of the points on the line.
Let’s use the point $P_2 = (4, 7)$. Substitute $x = 4$ and $y = 7$ into the equation:
$$
7 = 2 \times 4 + c
$$
So the y-intercept is $c = -1$.
Now substitute $m = 2$ and $c = -1$ back into:
$$y = mx + c$$
So, the equation of the line passing through the points $(2, 3)$ and $(4, 7)$ is:
$$
y = 2x\, – \,1
$$
How to Check If a Point Lies on a Line? (2:03)
Example 1: Check if the point $(3, 5)$ lies on the line $y = 2x\, – \,1$.
Substitute $x = 3$ into the equation:
$$y = 2 \times 3\, – \,1 = 5$$
Since $y = 5$, which matches the $y$-value of the point, the point $(3, 5)$ lies on the line $y = 2x\, – \,1$.
Example 2: Check if the point $(-2, -4)$ lies on the line $y = 2x\, – \,1$.
Substitute $x = -2$ into the equation:
$$y = 2 \times (-2)\, – \,1 = -5$$
But the point’s $y$-value is $-4$, not $-5$. So, the point $(-2, -4)$ is not on the line $y = 2x\, – \,1$.
📂 Flashcards: Finding Equations of a Straight Line
🍪 Quiz: Practice Finding the Equation of a Line
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