To find the equation of the line that passes through the points [tex]\((3, -4)\)[/tex] and [tex]\((-4, 3)\)[/tex], we need to determine its slope and y-intercept.
### Step 1: Calculate the Slope (m)
The formula for the slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given [tex]\((x_1, y_1) = (3, -4)\)[/tex] and [tex]\((x_2, y_2) = (-4, 3)\)[/tex], we substitute these coordinates into the formula:
[tex]\[ m = \frac{3 - (-4)}{-4 - 3} \][/tex]
[tex]\[ m = \frac{3 + 4}{-4 - 3} \][/tex]
[tex]\[ m = \frac{7}{-7} \][/tex]
[tex]\[ m = -1 \][/tex]
### Step 2: Determine the Y-intercept (b)
Using the slope-intercept form of a linear equation [tex]\( y = mx + b \)[/tex], we need to find the y-intercept [tex]\(b\)[/tex]. We already know [tex]\(m = -1\)[/tex]. To find [tex]\(b\)[/tex], we use one of the given points, say [tex]\((3, -4)\)[/tex].
Substitute [tex]\(x = 3\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(m = -1\)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[ -4 = (-1)(3) + b \][/tex]
[tex]\[ -4 = -3 + b \][/tex]
[tex]\[ b = -4 + 3 \][/tex]
[tex]\[ b = -1 \][/tex]
### Step 3: Write the Equation of the Line
Now that we have the slope [tex]\(m = -1\)[/tex] and the y-intercept [tex]\(b = -1\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = -1x - 1 \][/tex]
Simplifying the expression, we get:
[tex]\[ y = -x - 1 \][/tex]
Thus, the equation of the line that passes through the points [tex]\((3, -4)\)[/tex] and [tex]\((-4, 3)\)[/tex] is:
[tex]\[ \boxed{y = -x - 1} \][/tex]
