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?
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.
Hello, Haxcake!
You're expected to know how to solve a system of equations.
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 $\displaystyle (-5,\,1).$