1) Multiply the first equation by 7 and multiply the second equation by -7 to eliminate the x variable:
7(7x+3y) = 7(-25)
-7(-7x+9y) = -7(-47)
why is the first 7 positive and the second one is negative
To eliminate the x variable, we need to create opposing terms (one positive and one negative) so that when we add the two equations together, the x term cancels out. In this case, we multiply the first equation by 7 to get a positive 7x term, and we multiply the second equation by -7 to get a negative -7x term. By doing this, when we add the two equations together, the x terms will cancel each other out.
The first 7 is positive in order to multiply all terms inside the brackets by positive 7. This ensures that the signs of all terms in the equation remain the same.
On the other hand, the second 7 is negative in order to multiply all terms inside the brackets by negative 7. This will result in changing the signs of all terms in the equation. By doing this, when you add the two new equations together, the x terms will cancel out.
To eliminate the x variable in the system of equations, you need to multiply one equation by a certain number and the other equation by its negative counterpart. By doing so, when you add the two equations together, the x term will cancel out.
In this case, the first equation is multiplied by 7 and the second equation is multiplied by -7. The choice of positive 7 and negative 7 is arbitrary, and you can choose any nonzero number.
The reason why we choose positive 7 for the first equation and negative 7 for the second equation is because it allows us to create additive inverses for the x terms. When you multiply both terms of an equation by the same number, it does not change the equation's equality. The goal is to have the x terms of both equations to be additive inverses of each other, meaning they add up to zero when combined. By choosing one positive and one negative, we can achieve this result.
So, by multiplying the first equation by 7, we get:
7(7x + 3y) = 7(-25)
And by multiplying the second equation by -7, we get:
-7(-7x + 9y) = -7(-47)
Now, when you add these two equations together, the x term will cancel out.