Two lines with the slopes m1=4/3 and m2=-7/2 intersect at (3,4). Determine the equations of the two lines and check your answer by solving them
no clue how to answer it. I put each slope into the y=mx+b formula and found the b values, but i'm not sure if i'm supposed to do that
It helps if you show what you did.
The two lines, in point-slope form are
y-4 = 4/3 (x-3)
y-4 = -7/2 (x-3)
You don't always have to use y=mx+b. clearly, the two values for b are
0 and 29/2
With those, how could you not know what to do next? Knowing the slopes, you just write
y = 4/3 x
y = -7/2 x + 29/2
To determine the equations of the two lines, we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
where m represents the slope of the line, (x1, y1) represents a point on the line, and (x, y) represents any other point on the line.
Let's start with the first line, which has a slope of m1 = 4/3 and passes through the point (3, 4). Plugging these values into the point-slope form, we get:
y - 4 = (4/3)(x - 3)
To simplify, we can distribute the (4/3) to (x - 3):
y - 4 = (4/3)x - 4
Next, we can simplify further by adding 4 to both sides of the equation:
y = (4/3)x
So, the equation of the first line is y = (4/3)x.
Now, let's move on to the second line, which has a slope of m2 = -7/2 and passes through the point (3, 4). Substituting these values into the point-slope form, we have:
y - 4 = (-7/2)(x - 3)
To simplify, we can distribute the (-7/2) to (x - 3):
y - 4 = (-7/2)x + 21/2
Next, we can simplify further by adding 4 to both sides of the equation:
y = (-7/2)x + 21/2 + 4
Combining like terms on the right side:
y = (-7/2)x + 21/2 + 8/2
y = (-7/2)x + 29/2
So, the equation of the second line is y = (-7/2)x + 29/2.
To check our answer, we can solve these equations simultaneously. By setting the two equations equal to each other, we can find the x-coordinate of the point of intersection:
(4/3)x = (-7/2)x + 29/2
To simplify, we can multiply each term by 6 to eliminate the fractions:
8x = -21x + 87
Combining like terms:
29x = 87
Finally, dividing both sides by 29 gives us:
x = 3
Substituting this value of x back into either equation (e.g., the first one), we can find the corresponding y-coordinate:
y = (4/3)(3)
y = 4
Therefore, the point of intersection is (3, 4), which matches the given information.