Answer:
-3x and y are terms that cannot be combined, so their sum is -3x+y.
Step-by-step explanation:
You want an explanation of how y = 3x +4 becomes -3x +y = 4.
Balance
A balance is a mechanical contraption with two pans joined by a "beam" that rests on a pivot point between them. When the weights in the two pans are the same, the contraption is "balanced" and the pans sit level.
When the contraption is unbalanced, the heavier pan sits lower than the lighter one and the beam is not level.
The feature of the balance that is of interest here is that we can add or subtract weight from either pan. As long as we do the same thing to the other pan, it will remain balanced (the beam will be level).
Example
Consider a balance with yellow marbles and green marbles on both sides. Let the number of green marbles be 4, and the number of yellow marbles be 3.
If I remove 3 yellow marbles from the right side of the balance, so only the 4 green ones remain, I must also remove 3 yellow marbles from the left side to maintain the balance.
Equation
An equation is similar to a balance in that the parts on the left side and the parts on the right side of the equal sign have the same value when it is "true" or "balanced".
We can keep the statement of equality true as long as we perform the same operation on both sides of the equal sign.
The equation ...
y = 3x +4
says the value of y is the same as the value of the sum of 3x and 4. When I subtract (remove) 3x from the right side of this equation, it will only remain true (balanced) if I also remove 3x from the left side. Written on one line, this operation looks like ...
(-3x) +y = (-3x) +3x +4 . . . . . . -3x added to both sides
On the left side, the terms -3x and y have different variables. They are "unlike" terms, so cannot be combined. Their sum is "indicated" as ...
-3x +y
On the right side of the equal sign, the terms -3x and 3x have the same variable. They are "like" terms, so can be combined. The total number of instances of x in this sum is 0:
-3x + 3x = (-3 +3)x = 0x = 0
After the addition of -3x to both sides of the equation, the resulting equation is ...
-3x +y = 0 +4
We know that 0 is the additive identity element, so this can be further simplified to ...
-3x +y = 4
Bringing down
If we had the addition equation ...
70 + y = 30 +4
and we wanted to add -30 (same as subtract 30) to both sides, we could write this sum in a vertical format as ...
[tex]\begin{array}{rrcrr}70&+y&=&30&+4\\-30&&=&-30&\\\cline{1-5}40&+y&=&&+4\end{array}[/tex]
The sum in the left column is the sum of the like terms there. Since we started with something we could subtract 30 from, the resulting sum is a numerical value (40).
Your example seeks to show the same sort of operation, except that there are no like terms to add -3x to. The resulting sum of nothing (0) and -3x is -3x.
In the above example, if 70 were 0, the value on the bottom line in the first column would be -30. That is, the -30 would be "brought down", since it remains unchanged by addition to 0.
[tex]\begin{array}{rrcrr}0&+y&=&30&+4\\-30&&=&-30&\\\cline{1-5}-30&+y&=&&+4\end{array}[/tex]
Your example has -3x instead of -30. The ideas remain the same. Yes, the "-30" or "-3x" is "brought down", because there is nothing to add it to.
This vertical format for adding equations is often seen in a teaching setting. It is intended to show the equality of the stuff being added, and how it lines up with terms already there. When the vertical alignment is not preserved (as in your example), the comparison to the vertical alignment of columns in addition of multidigit numbers is lost. This tends to obscure the conceptual similarity.
Some may prefer the horizontal expression:
(-3x) +y = (-3x) +3x +4 . . . -3x added to both sides
-3x +y = 4 . . . . . . . . . . . . simplified
Standard form
As your example attempts to show, the standard form of the equation for a line is ...
Ax +By = C
The conditions on this form are generally ...
- the leading coefficient is positive
- the coefficients are mutually prime integers
Please note that the example you are presented shows addition of -3x to both sides of the equation. The property of equality also applies to multiplication: equality is preserved if both sides of the equation are multiplied by the same number.
In order to make the leading coefficient (the coefficient of x) be positive, we can multiply both sides of the equation by -1:
-1·(-3x +y) = -1·(4)
Parentheses are eliminated using the distributive property.
3x -y = -4 . . . . . . . . . the standard form of the equation for your line
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Additional comment
The standard form can be used for horizontal and vertical lines. In these cases, one of the coefficients is zero:
Ax = C . . . . . a vertical line at x = C/A
By = C . . . . . a horizontal line at y = C/B
The equation of a vertical line in slope-intercept form is impossible, because its slope is "undefined."
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