1/2x+1/3y=5 and 1/4x+y=10 solve this system by substitution

Mathematics

1 answer:

qaws [65]3 years ago

5 0

Answer:

(x, y) = (4, 9)

Step-by-step explanation:

You can use the second equation to write an expression for y.

y = 10 - 1/4x . . . . subtract 1/4x from the given 2nd equation

Now, substitute that for y in the first equation:

1/2x + 1/3(10 -1/4x) = 5

1/2x + 10/3 -1/12x = 5 . . . . eliminate parentheses

You can work this a couple of ways from here. One is to multiply by 12 to eliminate fractions. Another is to work with the numbers as they are. We'll do that, because we don't get enough practice doing arithmetic with fractions.

5/12x = 5/3 . . . . . subtract 10/3

x = (5/3)/(5/12) = 12/3 = 4

From above, we can use our expression for y:

y = 10 - 1/4·4 = 9

The solution is (x, y) = (4, 9).

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1. An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side. Find the equations of the altitudes of the triangle with vertices (4, 5),(-4, 1) and (2, -5). Do this by solving a system of two of two of the altitude equations and showing that the intersection point also belongs to the third line. 
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Let point
A=(xA,yA)
B=(xB,yB)
C=(xC,yC)

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Let's find the equation of the segment AB

Start with the general formula

Plug in the given points

Simplify and combine like terms

So the equation of the line through AB is

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Let's find the equation of the segment BC

Start with the general formula

Plug in the given points

Simplify and combine like terms

So the equation of the line through BC is

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Let's find the equation of the segment CA

Start with the general formula

Plug in the given points

Simplify and combine like terms

So the equation of the line through CA is

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So to find the equation of the line that is perpendicular to that goes through the point C(2,-5), simply negate and invert the slope to get

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Now plug the slope and the point (2,-5) into

Solve for y and simplify

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------------------------------------------------------------
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Plug in into the first equation

Add 2x to both sides and subtract 2 from both sides

Divide both sides by 3 to isolate x

Now plug this into

So the orthocenter is (-2/3,1/3)

So if we plug in into the third equation , we get

So the orthocenter lies on the third altitude

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