To write the equation of the line with a given slope and a point it passes through, we will use the point-slope form of the linear equation. The general formula for the point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where:
- [tex]\( m \)[/tex] is the slope
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line
Given:
- Slope [tex]\( m = 3 \)[/tex]
- Point [tex]\( (x_1, y_1) = (-5, -7) \)[/tex]
Let's substitute these values into the point-slope formula:
[tex]\[ y - (-7) = 3(x - (-5)) \][/tex]
Simplify the equation:
[tex]\[ y + 7 = 3(x + 5) \][/tex]
Next, distribute the slope on the right side:
[tex]\[ y + 7 = 3x + 15 \][/tex]
Now, solve for [tex]\( y \)[/tex] to get the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = 3x + 15 - 7 \][/tex]
Simplify the constants on the right-hand side:
[tex]\[ y = 3x + 8 \][/tex]
So, the equation of the line is:
[tex]\[ y = 3x + 8 \][/tex]
In addition to this, we can verify:
- The slope [tex]\( m = 3 \)[/tex]
- The given point [tex]\((-5, -7)\)[/tex]
- The intercept (constant term in the equation [tex]\( y = mx + b \)[/tex]) is [tex]\( 8 \)[/tex].
Thus, the detailed step-by-step solution results in the equation:
[tex]\[ y = 3x + 8 \][/tex]
