Why is the system of equations 7x+7y=42

Question

and 6x+y=36
equivalent to the system of equations x+y=6
and 6x+y=36
?(1 point)
Responses

The intercepts of the equations 7x+7y=42
and x+y=6
are (0,6)
and (6,0)
.
The intercepts of the equations 7 x plus 7 y equals 42 and x plus y equals 6 are left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis .

The equation 7x+7y=42
can be multiplied by 7 to produce the equivalent equation x+y=6
.
The equation 7 x plus 7 y equals 42 can be multiplied by 7 to produce the equivalent equation x plus y equals 6 .

The intercepts of the equations 7x+7y=42
and x+y=6
are (0,36)
and (6,0)
.
The intercepts of the equations 7 x plus 7 y equals 42 and x plus y equals 6 are left parenthesis 0 comma 36 right parenthesis and left parenthesis 6 comma 0 right parenthesis .

All equations share the point (6,0)
.

Answer 1

To understand why the system of equations \( 7x + 7y = 42 \) and \( 6x + y = 36 \) is equivalent to the system of equations \( x + y = 6 \) and \( 6x + y = 36 \), we can analyze the first equation.

Starting with the equation \( 7x + 7y = 42 \), we can simplify it by dividing every term by 7:

\[ \frac{7x}{7} + \frac{7y}{7} = \frac{42}{7} \]

This simplifies to:

\[ x + y = 6 \]

Now we have shown that \( 7x + 7y = 42 \) is equivalent to \( x + y = 6 \).

The second equation in both systems, \( 6x + y = 36 \), remains unchanged.

Thus, both systems actually contain the same linear equations, but expressed in different forms.

Among the provided options, the correct response is:

The equation \( 7x + 7y = 42 \) can be divided by 7 to produce the equivalent equation \( x + y = 6 \).