x+y=3,x=3y-5 & x-y=-5,y=-2x-4

Question

Solve the following systems of equations algebraically. Then graph each system to confirm your solution.

a) x+y=3          b) x-y=-5

    x=3y-5             y=-2x-4

Answer 1

I will do the first one for you.

 

x + y = 3              eq1

x = 3y - 5              eq2

 

 

We can substitute eq2 into eq1, since eq2 is already defined by x.

 

3y - 5 + y = 3

 

 

Solve for y.

 

4y = 8

 

  y = 2

 

 

Substitute this value of y into eq2 to solve for x.

 

x = 3(2) - 5

 

x = 6 - 5

 

x = 1

 

 

 

To solve graphically, we rearrange both equations so that they are in the  y=mx+b  form.

 

y = -x + 3               eq1

 

x + 5 = 3y     ---->   y = (1/3)x + (5/3)               eq2

 

 

Graph both lines on the same coordinate system.  To do this, plot the first point using the y-intercept and plot the second point using the slope.  Then connect the two points for each in a straight line.  You should have two lines.

 

For eq1, the two points will be  (0, 3) and (1, 2).  Connect these points throughout the entire coordinate system.

For eq2, the two points will be  (0, 5/3) and (3, 8/5).   Connect these points throughout the entire coordinate system.

 

 

Now that you have your lines, look for the point of intersection.  The point of intersection should be (1, 2).

 

Answer 2

You are going to want to use the substitution method for this one.

 

For part A, substitute in the x from the second line into the first line.  We know that x= 3y-5, so that first line looks like: (3y-5)+y=3.  Then, we combine the y's and the numbers:  3y+y-5=3.  Add 5 to both sides and we have: 4y=8.  Divide both sides by 4 and we get y=2.  Plug this back into the equation to find x.  x + 2= 3  and x=3(2)-5.  In either case, x=1. 

 

So, the solution for the part a is the point (1,2)

 

Do a similar trick for part b.

 

We know what y equals so we can substitute it into the first equation.

 

We get: x-(-2x-4)=-5

 

Remember, x-(-2x) is the same thing as x +2x.  So, we have 3x+4=-5.  We subtract 4 from each side and get 3x=-9.  Divide both sides by 3 and get x= -3

 

Next, we plug that back into the original equation to get y.  -3 -y =-5.  Add 3 to both sides and get -y=-2.  Divide by sides by -1 and we get y=2

 

So, the answer for part b is the point (-3, 2)