Mai wonders what would happen if we multiply equations. That is, we multiply the expressions on the left side of the two equations and set them equal to the expressions on the right side of the two equations.

Question

In system B, write out an equation that you would get if you multiply the two equations in this matter.

System B:

2x + y = 4
x - y = 11

Answer 1

To find the equation we get when multiplying the two equations in System B, we need to multiply the left sides of the two equations and set them equal to the right sides.

Multiplying the left sides:
(2x + y)(x - y)

Using the distributive property, we have:
2x^2 - 2xy + xy - y^2

Combining like terms:
2x^2 - xy - y^2

Setting this equal to the right sides of the equations:
2x^2 - xy - y^2 = 4 * 11

Simplifying further:
2x^2 - xy - y^2 = 44

So, the equation we get by multiplying the two equations in System B is:
2x^2 - xy - y^2 = 44

Answer 2

Can you further simplify this and solve it?

Answer 3

Unfortunately, the equation 2x^2 - xy - y^2 = 44 cannot be simplified any further. It is a quadratic equation in two variables (x and y) and cannot be completely solved unless we have more information or another equation to work with.

Answer 4

What if we used elimination?

Answer 5

To solve the system using elimination, we can multiply one or both equations by a constant to create coefficients that will cancel out when we add or subtract the equations.

Let's multiply the second equation by 2:
2(x - y) = 2(11)
2x - 2y = 22

Now, we can add the two equations:
(2x + y) + (2x - 2y) = 4 + 22

Simplifying:
4x - y = 26

Now we have a system of two equations:
2x + y = 4
4x - y = 26

To eliminate the y term, we can add the two equations:
(2x + y) + (4x - y) = 4 + 26

Simplifying:
6x = 30

Dividing both sides by 6:
x = 5

Now, substitute the value of x back into one of the original equations:
2(5) + y = 4
10 + y = 4
y = -6

Therefore, the solution to the system B is x = 5 and y = -6.