In the xy-plane, a system of inequalities consists of the following: y > -\frac{1}{2}x + 2… — SAT

In the $xy$ -plane, a system of inequalities consists of the following:

y > -\frac{1}{2}x + 2

y \leq 3x - 1

Which of the following coordinate pairs $(x, y)$ is a solution to this system?

The coordinate pair

(2, 2)

The coordinate pair

(2, 2)

is the correct answer because substituting

x = 2

and

y = 2

into both inequalities yields true statements. For the first inequality,

2 > -\frac{1}{2}(2) + 2

simplifies to

2 > 1

, which is true. For the second inequality,

2 \leq 3(2) - 1

simplifies to

2 \leq 5

, which is also true. Since the point satisfies both inequalities, it lies in the solution region.

Understand the definition of a solution to a system of inequalities.

A coordinate pair

(x, y)

is a solution to a system of inequalities if and only if it satisfies both inequalities simultaneously when substituted.

This establishes the verification method for checking the options.

Substitute the coordinate pair

(2, 2)

into the first inequality:

y > -\frac{1}{2}x + 2

2 > -\frac{1}{2}(2) + 2 \implies 2 > -1 + 2 \implies 2 > 1

This determines if the coordinate pair satisfies the first boundary condition.

Substitute the coordinate pair

(2, 2)

into the second inequality:

y \leq 3x - 1

2 \leq 3(2) - 1 \implies 2 \leq 6 - 1 \implies 2 \leq 5

This determines if the coordinate pair satisfies the second boundary condition.

Conclude whether both statements are true.

Since

2 > 1

is true and

2 \leq 5

is true, the coordinate pair

(2, 2)

is a solution to the system.

Both conditions must be met for the coordinate pair to belong to the solution set.

Verifying coordinate solutions for systems of linear inequalities

Estimated Time:1m 30s

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