Find the point of intersection of each pair of lines, if one exits. Check each solution if possible

A.

x=-2y-3

4y-x=9

B.

x+5y=8

-x+2y=-1

C.

4x-2y=5

y=2x+10

Can someone teach and show me how to solve these step by step?

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- Nov 8th 2006, 01:20 PMHaxcakeneed help! intersecting lines
Find the point of intersection of each pair of lines, if one exits. Check each solution if possible

A.

x=-2y-3

4y-x=9

B.

x+5y=8

-x+2y=-1

C.

4x-2y=5

y=2x+10

Can someone teach and show me how to solve these step by step? - Nov 8th 2006, 01:48 PMJameson
- Nov 8th 2006, 01:49 PMAfterShock

Finding the point of intersection for each of the pairs of lines is doing the following:

1.) Solving for y in each of the following

2.) Setting them equal to each other, since you just found that they both equal y. Solve for x and this will give you when they intersect at which x value.

Thus, they will intersect when they are equal.

For A:

x=-2y-3

y = (-x - 3)/2 (bring 2y to the left side and x to the right and divide by 2)

4y-x=9

y = (x+9)/4 (same method as above)

(-x - 3)/2 = (x+9)/4; solve for x.

Isolate x. Let's isolate x on the right side. Multiply the right side by 4, and thus you need to multiply the left by 4, too.

2*(-x-3) = x + 9

-2x - 6 = x + 9

Subtract 9 from right and left side and add +2x to both sides:

-15 = 3x

Divide by 3

x = -5; Thus, they intersect at x = -5; plug x = -5 into any of the equations and you will get the y value, too.

Let's pick the first equation: x=-2y-3.. so

-5 = -2y - 3;

-2 = -2y

y = 1;

Thus they intersect at (-5,1); you can also find this solution by graphing the two equations and seeing where they intersect on a graph.

For B and C, you do the exact same thing. For B I will give the answer, however, I think it is best that YOU try and do the work. If you get a conflicting answer, then show the work that you did and I can help you then. If you get the right answer, then perfect. No further explanation is needed.

B.) x = 3; y = 1, thus (3,1)

C is a little trickier:

C.) y = 2*x - 5/2

y=2x+10

Once you have gotten both in the "y = mx + b" form, you'll notice that the slope in both of the equations is 2; thus, they will never intersect. They have different y intersects so they will not coincide either. Graph those two functions and it will be obvious why they do not ever intersect. - Nov 8th 2006, 01:53 PMAfterShock
- Nov 8th 2006, 02:04 PMSoroban
Hello, Haxcake!

You're expected to know how to solve a system of equations.

Quote:

Find the point of intersection of each pair of lines, if one exists.

$\displaystyle A)\;\begin{array}{cc}(1)\\(2)\end{array} \begin{array}{cc}x\:=\:-2y-3 \\ 4y-x\:=\:9\end{array}$

This one suggests the Substitution Method.

. . (1) Solve one equation for one of its variables.

. . (2) Substitute this expression into the other equation.

. . (3) Solve the resulting equation for its variable.

. . (4) Solve for the other variable.

$\displaystyle \text{Equation (1) is already solved for }x:\;\;x\:=\:\underbrace{-2y -3}_\downarrow$

Substitute this into Equation (2): . . $\displaystyle 4y - (-2y - 3) \:=\:9$

And solve for $\displaystyle y:\;\;4y + 2y + 3 \:=\:9\quad\Rightarrow\quad 6y \:=\:6\quad\Rightarrow\quad\boxed{ y \,=\,1}$

Substitute this into Equation (1): .$\displaystyle x \:=\:-2(1) - 3\quad\Rightarrow\quad\boxed{ x\,=\,-5}$

Therefore, the intersection is $\displaystyle (-5,\,1).$