Problem 3 - College Algebra Essentials

Question

determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(2,5)\\\ &\left\\{\begin{aligned} 2 x+3 y &=17 \\ x+4 y &=16 \end{aligned}\right. \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pair (2,5) is not a solution to the system because it does not satisfy the second equation.

Step 1: Substitution in the first equation

Take the first equation in the system: \(2x + 3y = 17\), substitute x as 2 and y as 5: \(2(2) + 3(5) = 17\), which simplifies to: \(4 + 15 = 17\).

Step 2: Check the result of first equation

If the left hand side equals the right hand side then the equation is valid. In this case, both sides do equal 17, hence the first equation is satisfied.

Step 3: Substitution in the second equation

Take the second equation in the system: \(x + 4y = 16\), substitute x as 2 and y as 5: \(2 + 4(5) = 16\), which simplifies to: \(2 + 20 = 16\).

Step 4: Check the result of the second equation

If the left hand side equals the right hand side then the equation is valid. However, in this case, the left hand side equals 22, not 16, hence the ordered pair (2, 5) does not satisfy the second equation.

Key Concepts

Ordered Pairs

An ordered pair, like the one provided in the exercise, is a set of numbers used to represent a point in a specific space, typically a coordinate plane. An ordered pair is composed of two elements, the first usually denoting the x-coordinate and the second the y-coordinate.
For example, in the ordered pair (2, 5), '2' represents the x-value while '5' is the y-value. When trying to determine if an ordered pair is a solution to a system of equations, these values are substituted into each equation in the system to verify if they make all equations true simultaneously.
Ordered pairs are essential for solving systems of equations because they allow us to pinpoint specific solutions, suggesting where two linear equations intersect on a graph.

Substitution Method

The substitution method is a systematic process used to find if a specific solution satisfies a set of equations. It involves replacing a variable in an equation with a given value from an ordered pair.
In this exercise, we first substitute the x and y values from the ordered pair (2, 5), into each equation separately. For example, in the equation \(2x + 3y = 17\), we replace \(x\) with 2 and \(y\) with 5 to check if the equation holds true by calculating the expressions.
This method is particularly useful because it provides a clear, step-by-step approach to verifying solutions, reducing errors and ensuring clarity in complex calculations. The substitution method is not just limited to one type of problem but is widely applicable in mathematics wherever equations are involved.

Solution Validity

Determining the validity of a solution means checking whether the values satisfy all the equations involved in a system. In our current exercise, the steps involve plugging the ordered pair into both equations in the system.
In the first equation, \(2x + 3y = 17\), substituting the given x and y values results in the expression reducing correctly to 17, confirming that the choice satisfies this equation.
However, in the second equation, \(x + 4y = 16\), substituting produces \(2 + 20 = 22\), which does not equal 16. As a result, it proves that the ordered pair \((2, 5)\) is not a solution to the system because it does not satisfy both equations simultaneously.
This process highlights the importance of checking each equation in a system to ensure solution validity before concluding whether an ordered pair is correct.

Linear Equations

Linear equations form the backbone of algebra and are used extensively in various scientific fields. A linear equation is typically represented in the standard form \(ax + by = c\) where \(a\), \(b\), and \(c\) are constants.
These equations graph as straight lines on a coordinate plane. In a two-variable system, the solution to the linear equations corresponds to the intersection point of the two lines, which represents the set of values satisfying both equations simultaneously.
In our exercise, each equation represents a line. The challenge is to find a common ordered pair, if any, where these lines intersect. Linear equations have the unique feature that if a pair of values satisfies them, they form part of the infinite set of solutions that make the equtions true.
Understanding linear equations is essential as they provide foundational tools for problem-solving, analysis of patterns, and intersection studies in various quantitative fields.