Question

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{l}4 x^{2}+y=3 \\ -x-y=11\end{array}\right.\) (a) \((-2,-9)\) (b) \((2,-13)\)

Short answer

The pair (-2,-9) is not a solution to the system of equations, while the pair (2,-13) is a solution.

Step-by-step solution

Substitute and Test (a)

Put the first pair (-2,-9) into the equations. For the first equation: \(4(-2)^{2}+-9=3\) simplifies to \(16-9= 7\), which doesn’t equal to 3. So, the pair (-2,-9) doesn’t satisfy the first equation. We don't need to check the second equation since this ordered pair isn’t a solution to the system of equations.

Substitute and Test (b)

Put the second pair (2,-13) into the equations. For the first equation: \(4(2)^{2}-13=3\) simplifies to \(16-13= 3\), which matches with the right-hand side. Next, for the second equation: \(-2-(-13) = 11\), this simplifies to 11 which matches the right-hand side. So, the pair (2,-13) satisfies both equations, and thus it is a solution to the system of equations.

Key concepts

Solution Verification

Ensuring that an ordered pair is a solution for a system of equations involves validating if it satisfies all equations within that system. Here's how to do it:

• Substitution: First, substitute the values of the 'x' and 'y' from the ordered pair directly into each equation of the system.
• Check Equality: After substitution, perform the arithmetic operations to simplify both sides of each equation.
• Compare Results: The simplified left-hand side should be equal to the right-hand side of the equation. If it holds true, the ordered pair is a potential solution for that equation.

It's crucial to successfully pass this process for all equations in the system for the ordered pair to be a solution. If even one equation doesn't match, the pair isn't a solution. This steps bring clarity and assurance to whether the solution is correct, providing a solid understanding of the concept.

Algebraic Substitution

Algebraic substitution is a fundamental method in mathematics used to test solutions within equations. It helps in verifying potential solutions by working through a few simple steps:

• Plugging Values: Insert the 'x' and 'y' values from the ordered pair into the given equation.
• Calculation: Execute any required arithmetic such as squaring, multiplying, and adding or subtracting.
• Compare and Simplify: After calculation, compare the results to the constant term or right side of the equation to see if they match.

Substitution is particularly useful for systems of equations because it allows us to check fairly quickly if a given set of values solves the whole system. This technique is not limited to linear equations; it can also apply to quadratic and higher-degree equations.
The ease and straightforwardness of substitution make it an invaluable tool in problem-solving. Being comfortable with substitution helps build a strong mathematical foundation.

Ordered Pairs

Ordered pairs are a way to denote a solution in a two-variable context, typically represented as \((x, y)\). Here's how they function in systems of equations:

• Relation Representation: The 'x' and 'y' in an ordered pair represent specific values that might solve the equations when substituted.
• Graphical Interpretation: On a coordinate plane, these pairs indicate points where the two lines intersect, representing solutions to both equations simultaneously.
• Matching Equations: To verify if an ordered pair is a solution, both 'x' and 'y' must satisfy all equations in the system.

Understood correctly, ordered pairs are not just about numbers in parentheses. They represent possible solutions in a system, which can be analyzed both algebraically and graphically. This concept not only applies to basic algebra, but also expands into more complex systems, aiding in visualization and comprehension.