In Exercises 1–4, determine whether the given ordered pair is a solution of the system. (2, 5) 2x + 3y = 17 x + 4y = 16

Verified step by step guidance

1. Step 1: Understand that to determine if the ordered pair (2, 5) is a solution to the system, you need to substitute x = 2 and y = 5 into both equations and check if both equations hold true.
2. Step 2: Substitute x = 2 and y = 5 into the first equation: $\mathrm{2x}+\mathrm{3y}=\; 17$. This becomes $2(2)\; +\; 3(5)\; =\; ?\; 17$.
3. Step 3: Simplify the left side of the first equation: $4\; +\; 15\; =\; 19$. Compare this to the right side, which is 17. Since 19 ≠ 17, the first equation is not satisfied by the ordered pair.
4. Step 4: Substitute x = 2 and y = 5 into the second equation: $x+\; 4y=\; 16$. This becomes $2\; +\; 4(5)\; =\; ?\; 16$.
5. Step 5: Simplify the left side of the second equation: $2\; +\; 20\; =\; 22$. Compare this to the right side, which is 16. Since 22 ≠ 16, the second equation is also not satisfied by the ordered pair.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously. Understanding how to interpret and work with these systems is essential for solving problems involving multiple constraints.

Ordered Pair as a Solution

An ordered pair (x, y) represents a potential solution to a system of equations. To verify if it is a solution, substitute the values of x and y into each equation. If both equations are true after substitution, the ordered pair is a solution to the system.

Substitution and Verification Method

This method involves plugging the values of the ordered pair into each equation to check if the equations hold true. It is a straightforward way to determine if the given pair satisfies the system without solving the system from scratch.