Sure, let's solve the problem step-by-step.
The problem states that the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is 7, and that the value of [tex]\( y \)[/tex] is three more than the value of [tex]\( x \)[/tex].
To model this with a system of equations, let's break this down:
1. The sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is 7:
[tex]\[ x + y = 7 \][/tex]
2. The value of [tex]\( y \)[/tex] is three more than [tex]\( x \)[/tex]:
[tex]\[ y = x + 3 \][/tex]
Thus, the correct system of equations is:
[tex]\[ \left\{\begin{array}{l} x + y = 7 \\ y = x + 3 \end{array}\right. \][/tex]
Now, to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], follow these steps:
1. Substitute the second equation [tex]\( y = x + 3 \)[/tex] into the first equation [tex]\( x + y = 7 \)[/tex]:
[tex]\[ x + (x + 3) = 7 \][/tex]
2. Simplify the equation:
[tex]\[ x + x + 3 = 7 \][/tex]
[tex]\[ 2x + 3 = 7 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3 = 7 \][/tex]
[tex]\[ 2x = 7 - 3 \][/tex]
[tex]\[ 2x = 4 \][/tex]
[tex]\[ x = \frac{4}{2} \][/tex]
[tex]\[ x = 2 \][/tex]
4. Now that we have [tex]\( x \)[/tex], substitute [tex]\( x = 2 \)[/tex] back into the equation [tex]\( y = x + 3 \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[ y = 2 + 3 \][/tex]
[tex]\[ y = 5 \][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 5 \][/tex]
Therefore, the answer is:
[tex]\[ \left(2, 5\right) \][/tex]
