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Given the equation of the straight line AB is 3x+KY=8 where k is a constant. the straight line AB is parallel to the straight line connecting the point P(-1, 6) with the point Q(5, -3). Find the value of k then calculate the x-intercept of the straight line AB

Given The Equation Of The Straight Line AB Is 3xKY8 Where K Is A Constant The Straight Line AB Is Parallel To The Straight Line Connecting The Point P1 6 With T class=

Sagot :

If we have two parallel lines, than their slopes must be the same.

One way of comparing slopes of linear equations is to write them in the slope-intercept form:

[tex]y=mx+b[/tex]

In this form, m is the slope and b is the y-intercept.

So, let's write the first equation in this form:

[tex]\begin{gathered} 3x+ky=8 \\ ky=-3x+8 \\ y=-\frac{3}{k}x+\frac{8}{k} \end{gathered}[/tex]

To find the value of k, we can find the slope of the parallel line and compair it to the slope in this equation.

The slope of a line given two points on it can be calculated as:

[tex]m=\frac{y_2-y_1}{x_2-x_1}_{}[/tex]

Since we have points (-1, 6) and (5, -3), we can calculate the slope of the parallel line:

[tex]m=\frac{-3-6}{5-(-1)}=\frac{-9}{5+1}=-\frac{9}{6}=-\frac{3}{2}[/tex]

Since both lines are parallel, their slopes are the same.

We know that the slope of the first line is -3/k and the second line is -3/2, so, since they are parallel:

[tex]\begin{gathered} -\frac{3}{k}=-\frac{3}{2} \\ -3\cdot2=-3\cdot k \\ 2=k \\ k=2 \end{gathered}[/tex]

Since we have the value for k, we can substitute it into the equation for AB:

[tex]\begin{gathered} y=-\frac{3}{k}x+\frac{8}{k} \\ y=-\frac{3}{2}x+\frac{8}{2} \\ y=-\frac{3}{2}x+4 \end{gathered}[/tex]

To find the x-intercept, we can see that it happens when the value of y is equal to 0, so we can plug in y = 0 and find the value of x:

[tex]\begin{gathered} y=0 \\ 0=-\frac{3}{2}x+4 \\ \frac{3}{2}x=4 \\ x=\frac{2}{3}\cdot4 \\ x=\frac{8}{3} \end{gathered}[/tex]

So, the value of k is 2 and the x-intercept is 8/3.

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