Question
Why is the system of equations and equivalent to the system
of equations and ? (1 point)
7x + 7y = 42 6x + y = 36
x + y = 6 6x + y = 36
The intercepts of the equations 7x + 7y = 42 and x + y = 6 are (0, 36) and (6, 0).
The intercepts of the equations 7x + 7y = 42 and x + y = 6 are (0, 6) and (6, 0).
The equation can be multiplied by 7 to produce the equivalent equation
.
7x + 7y = 42
x + y = 6
All equations share the point (6, 0)
of equations and ? (1 point)
7x + 7y = 42 6x + y = 36
x + y = 6 6x + y = 36
The intercepts of the equations 7x + 7y = 42 and x + y = 6 are (0, 36) and (6, 0).
The intercepts of the equations 7x + 7y = 42 and x + y = 6 are (0, 6) and (6, 0).
The equation can be multiplied by 7 to produce the equivalent equation
.
7x + 7y = 42
x + y = 6
All equations share the point (6, 0)
Answers
GPT-4o mini
To determine why the two systems of equations are equivalent, we need to analyze the provided equations and their properties.
1. The equations provided are:
- \( 7x + 7y = 42 \)
- \( x + y = 6 \)
- \( 6x + y = 36 \)
2. First, let's simplify \( 7x + 7y = 42 \):
- Dividing the entire equation by 7 gives:
\[
x + y = 6
\]
This shows that \( 7x + 7y = 42 \) is equivalent to \( x + y = 6 \).
3. Next, let's analyze the equation \( 6x + y = 36 \):
- We can subtract \( 6x \) from both sides to express \( y \):
\[
y = 36 - 6x
\]
This can be expressed in different forms but will still describe the same line in the coordinate system.
4. By substituting the equivalent equation \( x + y = 6 \) into \( 6x + y = 36 \), we can conclude if these two systems have any common points of intersection:
- From \( x + y = 6 \), we can express \( y \) in terms of \( x \):
\[
y = 6 - x
\]
- Substituting \( y \) into \( 6x + y = 36 \):
\[
6x + (6 - x) = 36
\]
Simplifying gives:
\[
5x + 6 = 36 \implies 5x = 30 \implies x = 6
\]
- If \( x = 6 \), substituting back for \( y \) gives:
\[
y = 6 - 6 = 0
\]
So, the intersection point is \( (6, 0) \).
5. The intercepts for the equations are:
- For \( 7x + 7y = 42 \):
- \( x \)-intercept (set \( y = 0 \)): \( 7x = 42 \implies x = 6 \) so \( (6, 0) \)
- \( y \)-intercept (set \( x = 0 \)): \( 7y = 42 \implies y = 6 \) so \( (0, 6) \)
- For \( x + y = 6 \):
- \( x \)-intercept (set \( y = 0 \)): \( x = 6 \) gives \( (6, 0) \)
- \( y \)-intercept (set \( x = 0 \)): \( y = 6 \) gives \( (0, 6) \)
- The equation \( 6x + y = 36 \):
- \( x \)-intercept (set \( y = 0 \)): \( 6x = 36 \implies x = 6 \) gives \( (6, 0) \)
- \( y \)-intercept (set \( x = 0 \)): \( y = 36 \) gives \( (0, 36) \)
From the conclusions reached above, we can verify that:
- The systems \( 7x + 7y = 42\) and \( x + y = 6\) are equivalent because they represent the same line.
- The equations have the common point of \( (6, 0) \), which validates them being equivalent.
- The intercepts for \( x + y = 6\) are correctly given as \( (6, 0) \) and \( (0, 6) \), but this was mistakenly noted in the interpretations.
Thus, the statement "The equation can be multiplied by 7 to produce the equivalent equation" is correct, and "All equations share the point (6, 0)" is also correct, affirming equivalence.
1. The equations provided are:
- \( 7x + 7y = 42 \)
- \( x + y = 6 \)
- \( 6x + y = 36 \)
2. First, let's simplify \( 7x + 7y = 42 \):
- Dividing the entire equation by 7 gives:
\[
x + y = 6
\]
This shows that \( 7x + 7y = 42 \) is equivalent to \( x + y = 6 \).
3. Next, let's analyze the equation \( 6x + y = 36 \):
- We can subtract \( 6x \) from both sides to express \( y \):
\[
y = 36 - 6x
\]
This can be expressed in different forms but will still describe the same line in the coordinate system.
4. By substituting the equivalent equation \( x + y = 6 \) into \( 6x + y = 36 \), we can conclude if these two systems have any common points of intersection:
- From \( x + y = 6 \), we can express \( y \) in terms of \( x \):
\[
y = 6 - x
\]
- Substituting \( y \) into \( 6x + y = 36 \):
\[
6x + (6 - x) = 36
\]
Simplifying gives:
\[
5x + 6 = 36 \implies 5x = 30 \implies x = 6
\]
- If \( x = 6 \), substituting back for \( y \) gives:
\[
y = 6 - 6 = 0
\]
So, the intersection point is \( (6, 0) \).
5. The intercepts for the equations are:
- For \( 7x + 7y = 42 \):
- \( x \)-intercept (set \( y = 0 \)): \( 7x = 42 \implies x = 6 \) so \( (6, 0) \)
- \( y \)-intercept (set \( x = 0 \)): \( 7y = 42 \implies y = 6 \) so \( (0, 6) \)
- For \( x + y = 6 \):
- \( x \)-intercept (set \( y = 0 \)): \( x = 6 \) gives \( (6, 0) \)
- \( y \)-intercept (set \( x = 0 \)): \( y = 6 \) gives \( (0, 6) \)
- The equation \( 6x + y = 36 \):
- \( x \)-intercept (set \( y = 0 \)): \( 6x = 36 \implies x = 6 \) gives \( (6, 0) \)
- \( y \)-intercept (set \( x = 0 \)): \( y = 36 \) gives \( (0, 36) \)
From the conclusions reached above, we can verify that:
- The systems \( 7x + 7y = 42\) and \( x + y = 6\) are equivalent because they represent the same line.
- The equations have the common point of \( (6, 0) \), which validates them being equivalent.
- The intercepts for \( x + y = 6\) are correctly given as \( (6, 0) \) and \( (0, 6) \), but this was mistakenly noted in the interpretations.
Thus, the statement "The equation can be multiplied by 7 to produce the equivalent equation" is correct, and "All equations share the point (6, 0)" is also correct, affirming equivalence.