Question
Why is the system of equations 2x+7y=14
2
𝑥
+
7
𝑦
=
14
and 4x+5y=10
4
𝑥
+
5
𝑦
=
10
equivalent to the system of equations 4x+14y=28
4
𝑥
+
14
𝑦
=
28
and 8x+10y=20
8
𝑥
+
10
𝑦
=
20
?(1 point)
Responses
Both equations, 2x+7y=14
2
𝑥
+
7
𝑦
=
14
and 4x+5y=10
4
𝑥
+
5
𝑦
=
10
, can be multiplied by 12
1
2
to produce the equivalent system of equations.
Both equations, 2 x plus 7 y equals 14 and 4 x plus 5 y equals 10 , can be multiplied by Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations.
All equations share the point (0,2)
(
0
,
2
)
.
All equations share the point left parenthesis 0 comma 2 right parenthesis .
The intercepts of both 4x+5y=10
4
𝑥
+
5
𝑦
=
10
and 8x+10y=20
8
𝑥
+
10
𝑦
=
20
are (0,2.5)
(
0
,
2.5
)
and (2,0)
(
2
,
0
)
.
The intercepts of both 4x+5y=10
4
𝑥
+
5
𝑦
=
10
and 8x+10y=20
8
𝑥
+
10
𝑦
=
20
are (0,2.5)
(
0
,
2.5
)
and (2,0)
(
2
,
0
)
.
The intercepts of both 4 x plus 5 y equals 10 and 8 x plus 10 y equals 20 are left parenthesis 0 comma 2.5 right parenthesis and left parenthesis 2 comma 0 right parenthesis .
The intercepts of the equations 2x+7y=14
2
𝑥
+
7
𝑦
=
14
and 4x+14y=28
4
𝑥
+
14
𝑦
=
28
are (0,2)
(
0
,
2
)
and (7,0)
(
7
,
0
)
.
2
𝑥
+
7
𝑦
=
14
and 4x+5y=10
4
𝑥
+
5
𝑦
=
10
equivalent to the system of equations 4x+14y=28
4
𝑥
+
14
𝑦
=
28
and 8x+10y=20
8
𝑥
+
10
𝑦
=
20
?(1 point)
Responses
Both equations, 2x+7y=14
2
𝑥
+
7
𝑦
=
14
and 4x+5y=10
4
𝑥
+
5
𝑦
=
10
, can be multiplied by 12
1
2
to produce the equivalent system of equations.
Both equations, 2 x plus 7 y equals 14 and 4 x plus 5 y equals 10 , can be multiplied by Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations.
All equations share the point (0,2)
(
0
,
2
)
.
All equations share the point left parenthesis 0 comma 2 right parenthesis .
The intercepts of both 4x+5y=10
4
𝑥
+
5
𝑦
=
10
and 8x+10y=20
8
𝑥
+
10
𝑦
=
20
are (0,2.5)
(
0
,
2.5
)
and (2,0)
(
2
,
0
)
.
The intercepts of both 4x+5y=10
4
𝑥
+
5
𝑦
=
10
and 8x+10y=20
8
𝑥
+
10
𝑦
=
20
are (0,2.5)
(
0
,
2.5
)
and (2,0)
(
2
,
0
)
.
The intercepts of both 4 x plus 5 y equals 10 and 8 x plus 10 y equals 20 are left parenthesis 0 comma 2.5 right parenthesis and left parenthesis 2 comma 0 right parenthesis .
The intercepts of the equations 2x+7y=14
2
𝑥
+
7
𝑦
=
14
and 4x+14y=28
4
𝑥
+
14
𝑦
=
28
are (0,2)
(
0
,
2
)
and (7,0)
(
7
,
0
)
.
Answers
Answered by
GPT-4o mini
The correct reason why the system of equations
1. \( 2x + 7y = 14 \)
2. \( 4x + 5y = 10 \)
is equivalent to the system of equations
3. \( 4x + 14y = 28 \)
4. \( 8x + 10y = 20 \)
is because you can multiply the equations by a specific factor to get equivalent forms:
- The first equation \( 2x + 7y = 14 \) can be multiplied by 2 to yield \( 4x + 14y = 28 \).
- The second equation \( 4x + 5y = 10 \) can be multiplied by 2 to yield \( 8x + 10y = 20 \).
This means that the equations are scaled versions of each other, maintaining the same solutions.
So, the first response correctly states:
**Both equations, \( 2x + 7y = 14 \) and \( 4x + 5y = 10 \), can be multiplied by 2 to produce the equivalent system of equations.**
1. \( 2x + 7y = 14 \)
2. \( 4x + 5y = 10 \)
is equivalent to the system of equations
3. \( 4x + 14y = 28 \)
4. \( 8x + 10y = 20 \)
is because you can multiply the equations by a specific factor to get equivalent forms:
- The first equation \( 2x + 7y = 14 \) can be multiplied by 2 to yield \( 4x + 14y = 28 \).
- The second equation \( 4x + 5y = 10 \) can be multiplied by 2 to yield \( 8x + 10y = 20 \).
This means that the equations are scaled versions of each other, maintaining the same solutions.
So, the first response correctly states:
**Both equations, \( 2x + 7y = 14 \) and \( 4x + 5y = 10 \), can be multiplied by 2 to produce the equivalent system of equations.**