Math

Question Determine if the point (2,1)(2,1) satisfies the system of inequalities 5x+2y<185x + 2y < 18 and 5x+y>15x + y > 1.

Studdy Solution

STEP 1

Assumptions
1. We have a system of two inequalities: 5x+2y<185x + 2y < 18 5x+y>15x + y > 1
2. We need to check if the point (2,1)(2,1) is a solution to the system.
3. A point is a solution to a system of inequalities if it satisfies all the inequalities when its coordinates are substituted into them.

STEP 2

First, we substitute the x-coordinate of the point (2,1)(2,1) into the first inequality.
5x+2y<185x + 2y < 18

STEP 3

Perform the substitution with x=2x = 2 and y=1y = 1.
5(2)+2(1)<185(2) + 2(1) < 18

STEP 4

Calculate the left-hand side of the inequality.
10+2<1810 + 2 < 18

STEP 5

Simplify the left-hand side of the inequality.
12<1812 < 18

STEP 6

Since 1212 is less than 1818, the point (2,1)(2,1) satisfies the first inequality.

STEP 7

Next, we substitute the x-coordinate and y-coordinate of the point (2,1)(2,1) into the second inequality.
5x+y>15x + y > 1

STEP 8

Perform the substitution with x=2x = 2 and y=1y = 1.
5(2)+1>15(2) + 1 > 1

STEP 9

Calculate the left-hand side of the inequality.
10+1>110 + 1 > 1

STEP 10

Simplify the left-hand side of the inequality.
11>111 > 1

STEP 11

Since 1111 is greater than 11, the point (2,1)(2,1) also satisfies the second inequality.

STEP 12

Because the point (2,1)(2,1) satisfies both inequalities, it is a solution to the system of inequalities.
The answer is "yes".

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