a linear function has the same y-intercept as x+4y=12 and its graph contains the point(5.4).find the y-intercept and the slope of the linear function
The y-intercept of the line
L1 : x+4y=12 can be found by putting the equation in slope-intercept form,
L1 : y = -(1/4)x + 3
The intercept is thus +3, and the point of intercept is (0,3).
The linear function therefore passes through the points P1(0,3) and P2(5,4).
The slope m,can be found by
m=(y2-y1)/(x2-x1) as long as x2≠x1.
The line passing through the two given points is
L2 : (y-y1)/(y2-y1)=(x-x1)/(x2-x1)
where y2≠y1 and x2≠x1.
To find the y-intercept and slope of a linear function, we need to first rewrite the given equation in the standard form, which is y = mx + b, where m is the slope and b is the y-intercept.
Let's start by rearranging the given equation x + 4y = 12. First, subtract x from both sides of the equation:
4y = -x + 12.
Next, divide both sides of the equation by 4 to isolate y:
y = (-1/4)x + 3.
Now, we have the equation in the standard form, y = mx + b. Comparing this equation to the standard form, we can see that the slope of the linear function is -1/4.
To find the y-intercept, we need to substitute the values of the x and y coordinates of the given point (5, 4) into the equation y = (-1/4)x + b. Let's do that:
4 = (-1/4)(5) + b.
Simplifying the equation:
4 = (-5/4) + b.
To isolate b, we need to add 5/4 to both sides of the equation:
4 + 5/4 = b.
Multiplying 4 by 4/4 to get the same denominator:
16/4 + 5/4 = b.
Simplifying the left side of the equation:
21/4 = b.
So, the y-intercept of the linear function is 21/4 (or 5.25).
Therefore, the slope of the linear function is -1/4, and the y-intercept is 21/4 (or 5.25).