Let's break down the word problem step-by-step to find the system of equations that correctly represents it.
The problem states:
1. The larger of two numbers exceeds twice the smaller by one.
- Let [tex]\( y \)[/tex] be the larger number.
- Let [tex]\( x \)[/tex] be the smaller number.
We can represent this as:
[tex]\[
y = 2x + 1
\][/tex]
2. Three times the smaller exceeds the larger by six.
- This means three times [tex]\( x \)[/tex] exceeds [tex]\( y \)[/tex] by 6.
We can represent this as:
[tex]\[
3x - y = 6
\][/tex]
Combining these two pieces of information, our system of equations becomes:
[tex]\[
y = 2x + 1
\][/tex]
[tex]\[
3x - y = 6
\][/tex]
Now, let's compare these equations with the given options:
1. [tex]\( y = 2(x + 1) \)[/tex] and [tex]\( y + 6 = 3x \)[/tex]
- [tex]\( y = 2(x + 1) \)[/tex] simplifies to [tex]\( y = 2x + 2 \)[/tex], which is not the same as [tex]\( y = 2x + 1 \)[/tex].
- [tex]\( y + 6 = 3x \)[/tex] simplifies to [tex]\( y = 3x - 6 \)[/tex], which is equivalent to [tex]\( 3x - y = 6 \)[/tex].
However, [tex]\( y = 2(x+1) \)[/tex] is incorrect because it simplifies to [tex]\( y = 2x + 2 \)[/tex].
2. [tex]\( y = 2x + 1 \)[/tex] and [tex]\( 3x + 6 = y \)[/tex]
- The equation [tex]\( y = 2x + 1 \)[/tex] is correct.
- The equation [tex]\( 3x + 6 = y \)[/tex] simplifies to [tex]\( y = 3x + 6 \)[/tex], which is incorrect. We need [tex]\( 3x - y = 6 \)[/tex].
3. [tex]\( y = 2x + 1 \)[/tex] and [tex]\( y + 6 = 3x \)[/tex]
- The equation [tex]\( y = 2x + 1 \)[/tex] is correct.
- The equation [tex]\( y + 6 = 3x \)[/tex] simplifies to [tex]\( 3x - y = 6 \)[/tex], which is also correct.
Therefore, the correct system of equations that represents the word problem is:
[tex]\[
y = 2x + 1 \quad \text{and} \quad y + 6 = 3x
\][/tex]
The correct option is:
[tex]\[
y = 2x + 1 \quad \text{and} \quad y + 6 = 3x
\][/tex]
