To find the equation of a line passing through the points [tex]\((-5, -3)\)[/tex] and [tex]\( (2, 1)\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex], follow these steps:
### Step 1: Calculate the Slope (m)
The slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's substitute the values [tex]\((-5, -3)\)[/tex] and [tex]\( (2, 1)\)[/tex]:
[tex]\[ m = \frac{1 - (-3)}{2 - (-5)} \][/tex]
[tex]\[ m = \frac{1 + 3}{2 + 5} \][/tex]
[tex]\[ m = \frac{4}{7} \][/tex]
So, the slope [tex]\( m \)[/tex] is:
[tex]\[ m = 0.5714285714285714 \][/tex]
### Step 2: Calculate the Y-Intercept (b)
The slope-intercept form of the equation is [tex]\( y = mx + b \)[/tex]. To find the value of [tex]\(b\)[/tex], use one of the points and solve for [tex]\( b \)[/tex]. Let's use the point [tex]\((-5, -3)\)[/tex].
Substitute [tex]\( m \)[/tex], [tex]\( x_1 \)[/tex], and [tex]\( y_1 \)[/tex]:
[tex]\[ y_1 = m x_1 + b \][/tex]
[tex]\[ -3 = (0.5714285714285714)(-5) + b \][/tex]
Simplify to solve for [tex]\( b \)[/tex]:
[tex]\[ -3 = -2.8571428571428572 + b \][/tex]
[tex]\[ b = -3 + 2.8571428571428572 \][/tex]
[tex]\[ b = -0.14285714285714324 \][/tex]
### Step 3: Write the Equation
Now that we have the slope [tex]\( m = 0.5714285714285714 \)[/tex] and the y-intercept [tex]\( b = -0.14285714285714324 \)[/tex], the equation of the line in slope-intercept form is:
[tex]\[ y = 0.5714285714285714x - 0.14285714285714324 \][/tex]
Therefore, the equation of the line passing through the points [tex]\((-5, -3)\)[/tex] and [tex]\( (2, 1)\)[/tex] is:
[tex]\[ y = 0.5714285714285714x - 0.14285714285714324 \][/tex]
