Sure, let's go through the detailed, step-by-step solution to find the slope-intercept form of a linear equation.
### Given:
The general slope-intercept equation is noted as:
[tex]\[ y = mx + b \][/tex]
Where:
- \(m\) is the slope of the line.
- \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).
### Step-by-Step Solution:
1. Understand the Point-Slope Form:
The point-slope form of a linear equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, \((x_1, y_1)\) is a point on the line.
2. Rearranging to Find the Slope-Intercept Form:
To get the slope-intercept form from the point-slope form, we need to isolate \(y\) on one side:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Add \(y_1\) to both sides:
[tex]\[ y = m(x - x_1) + y_1 \][/tex]
Distribute \(m\) on the right-hand side:
[tex]\[ y = mx - mx_1 + y_1 \][/tex]
3. Identify the Y-Intercept, \(b\):
The expression \(-mx_1 + y_1\) can be treated as a single constant since it doesn't depend on \(x\). This constant is the y-intercept \(b\):
[tex]\[ b = -mx_1 + y_1 \][/tex]
4. Resulting Slope-Intercept Equation:
Substitute \(b\) back into the equation:
[tex]\[ y = mx + b \][/tex]
Thus, after following these steps, we understand that the slope-intercept equation of a line is:
[tex]\[ y = mx + b \][/tex]
This form shows the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in terms of the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex].
