Sure, let's work through this problem step-by-step to find the equation of the line.
### 1. Find the Intersection Point
First, we need to find the intersection of the two given lines:
1. [tex]\( x + y - 4 = 0 \)[/tex]
2. [tex]\( 3x - 4y - 5 = 0 \)[/tex]
We can solve these two equations simultaneously to find the intersection point.
From the first equation:
[tex]\[ x + y = 4 \][/tex]
[tex]\[ y = 4 - x \][/tex]
Substitute [tex]\( y = 4 - x \)[/tex] into the second equation:
[tex]\[ 3x - 4(4 - x) - 5 = 0 \][/tex]
[tex]\[ 3x - 16 + 4x - 5 = 0 \][/tex]
[tex]\[ 7x - 21 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] back into [tex]\( y = 4 - x \)[/tex]:
[tex]\[ y = 4 - 3 = 1 \][/tex]
Therefore, the intersection point is [tex]\( (3, 1) \)[/tex].
### 2. Find the Slope of the Perpendicular Line
Next, we need to find the slope of the line that is perpendicular to the given line:
[tex]\[ 5x - y + 4 = 0 \][/tex]
First, we rearrange this equation to slope-intercept form [tex]\( y = mx + c \)[/tex]:
[tex]\[ 5x - y + 4 = 0 \][/tex]
[tex]\[ 5x + 4 = y \][/tex]
[tex]\[ y = 5x + 4 \][/tex]
The slope [tex]\( m \)[/tex] of this line is 5. The slope of the line perpendicular to this one is the negative reciprocal of 5, which is:
[tex]\[ -\frac{1}{5} \][/tex]
### 3. Use the Point-Slope Form to Find the Perpendicular Line
The equation of a line through a point [tex]\( (x_0, y_0) \)[/tex] with a given slope [tex]\( m \)[/tex] can be written using the point-slope form:
[tex]\[ y - y_0 = m(x - x_0) \][/tex]
In our case, the point is [tex]\( (3, 1) \)[/tex] and the slope is [tex]\( -\frac{1}{5} \)[/tex]. Plugging these values into the point-slope form, we get:
[tex]\[ y - 1 = -\frac{1}{5}(x - 3) \][/tex]
Simplify this equation to get it in the slope-intercept form:
[tex]\[ y - 1 = -\frac{1}{5}x + \frac{3}{5} \][/tex]
[tex]\[ y = -\frac{1}{5}x + \frac{3}{5} + 1 \][/tex]
[tex]\[ y = -\frac{1}{5}x + \frac{3}{5} + \frac{5}{5} \][/tex]
[tex]\[ y = -\frac{1}{5}x + \frac{8}{5} \][/tex]
Hence, the equation of the line through the intersection of [tex]\( x + y - 4 = 0 \)[/tex] and [tex]\( 3x - 4y - 5 = 0 \)[/tex], and perpendicular to [tex]\( 5x - y + 4 = 0 \)[/tex] is:
[tex]\[ y = -\frac{1}{5}x + \frac{8}{5} \][/tex]
Alternatively, it can also be expressed as:
[tex]\[ y - 1 = -\frac{1}{5}(x - 3) \][/tex]
