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.
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
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
Can you further simplify this and solve it?
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.
What if we used elimination?
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.
To multiply the two equations in System B, we will multiply the left side of the first equation by the left side of the second equation and set it equal to the product of the right sides of the equations.
Starting with the first equation: (2x + y)
And the second equation: (x - y)
Multiplying them together, we get:
(2x + y) * (x - y) = (4) * (11)
Expanding the left side of the equation, we have:
2x^2 - 2xy + xy - y^2 = 44
Combining like terms, we get:
2x^2 - xy - y^2 = 44
Therefore, the equation we get by multiplying the two equations in this manner is:
2x^2 - xy - y^2 = 44
To multiply the two equations in System B, we would multiply the expressions on the left side of the equations and set them equal to the expressions on the right side. Therefore, we can multiply the left sides of the equations:
(2x + y) * (x - y)
To simplify this expression, we can use the distributive property to multiply each term on the left side by each term on the right side:
2x * x - 2x * y + y * x - y * y
This simplifies to:
2x^2 - 2xy + xy - y^2
Combining like terms, we get:
2x^2 - xy - y^2
Therefore, the equation we would get by multiplying the two equations in System B is:
2x^2 - xy - y^2 = (4)(11) = 44