System of equations with substitution (algebraic) I need a lot of help with

Question

Solve the system of equations using subtractions. Show all work!

#1 y=x+3 3y+x=5

#2 x=-2y+9 -2x+3y=10

and using desmos graphing calc. I already figured out that for #1 X=-1 and Y=2 and so the solution is (-1,2) for and for #2 X=1 and Y=4 and the solution = (1,4)

but I need to know how to do it without using desmos, can you show me it step by step please, like the formula and stuff because I don't understand how to do it

Answer 1

You want to do it by subtraction.

#1 y=x+3 3y+x=5

First, rearrange the equations so that the x's and y's are on one side of the equation,

y=x+3 becomes y - x = 3 3y + x = 5 stays as is.

Put one equation above the other so that the y's and x's are aligned under each other,

y - x = 3

3y + x = 5

Now add the y's, x's and numbers algebraically.

4y + 0 = 8 solve for y to get y = 8/4 = 2

Now substitute the result for y into either equation.

3 * (2) + x = 5 solve for x to get x = -1

#2 x=-2y+9 -2x+3y=10

To do this by subtraction requires one extra step than above.

Rearrange to get x +2y = 9 and leave the 2nd equation as is and put one over the other.

x + 2y = 9

-2x +3y = 10

Adding as is won't result in any cancellations, so multiply the first equation through by 2 to get

2x + 4y = 18

-2x +3y = 10

Now add the two equations. The x's cancel to get

7y = 28

y = 4 substitute into either equation

2x + 4*(4) = 18 solve for x

2x = 2 therefore x = 1

I know this is lengthy, but I wanted to show you step-by-step.

Answer 2

1) 3(x+3)+x=5

3x+9+x=5

4x+9=5

4x = -4

x = -1

y = -1+3=2

x=-1 y=2

2) -2(-2y+9) + 3y = 10

4y-18 + 3y=10

7y -18 = 10

7y = 28

y = 4

x = -2*4+9

= -8+9

= 1

(x=1,y=4)