To determine the equation of a straight line given its gradient (slope) and a point through which it passes, we can follow these steps:
1. Identify the given information:
- Gradient (slope), [tex]\( m = -5 \)[/tex]
- Point through which the line passes, [tex]\( (x_1, y_1) = (13, 5) \)[/tex]
2. Use the point-slope form of the equation of a line:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the given values [tex]\( m = -5 \)[/tex], [tex]\( x_1 = 13 \)[/tex], and [tex]\( y_1 = 5 \)[/tex], we get:
[tex]\[
y - 5 = -5(x - 13)
\][/tex]
3. Simplify the equation to the slope-intercept form:
The slope-intercept form is [tex]\( y = mx + c \)[/tex]. We need to rearrange and simplify the point-slope equation:
[tex]\[
y - 5 = -5(x - 13)
\][/tex]
Distribute the [tex]\(-5\)[/tex] on the right-hand side:
[tex]\[
y - 5 = -5x + 65
\][/tex]
Add 5 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -5x + 70
\][/tex]
4. Identify the slope and the y-intercept:
In the equation [tex]\( y = -5x + 70 \)[/tex], the slope ([tex]\( m \)[/tex]) is [tex]\(-5\)[/tex] and the y-intercept ([tex]\( c \)[/tex]) is [tex]\( 70 \)[/tex].
So, the equation of the straight line with a gradient of [tex]\(-5\)[/tex] that passes through the point [tex]\((13, 5)\)[/tex] is:
[tex]\[
y = -5x + 70
\][/tex]
