To determine if the ordered pair (3, 10) is a solution to the given system of inequalities, we will substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 10 \)[/tex] into each inequality and check if both are satisfied.
The given system of inequalities is:
1. [tex]\( y > 2x + 1 \)[/tex]
2. [tex]\( y < 3x + 7 \)[/tex]
Step 1: Check the first inequality [tex]\( y > 2x + 1 \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 10 \)[/tex] into the inequality:
[tex]\[
10 > 2(3) + 1
\][/tex]
Calculate the right-hand side:
[tex]\[
10 > 6 + 1
\][/tex]
[tex]\[
10 > 7
\][/tex]
This inequality is true.
Step 2: Check the second inequality [tex]\( y < 3x + 7 \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 10 \)[/tex] into the inequality:
[tex]\[
10 < 3(3) + 7
\][/tex]
Calculate the right-hand side:
[tex]\[
10 < 9 + 7
\][/tex]
[tex]\[
10 < 16
\][/tex]
This inequality is also true.
Since both inequalities are satisfied, the ordered pair [tex]\( (3, 10) \)[/tex] is a solution to the given system of inequalities.
Therefore, the answer is:
Yes, (3, 10) is a solution of the given system.
