Problem 2 - College Algebra with Modeling and Visualization

Question

Does the ordered triple \((1,2,3)\) satisfy the equation \(3 x+2 y+z=10 ?\)

Step-by-Step Solution

Answer

Yes, the ordered triple (1,2,3) satisfies the equation.

Step 1: Substitute Values into the Equation

The ordered triple is \((x, y, z) = (1, 2, 3)\). We need to substitute these values into the given equation \(3x + 2y + z = 10\). Substitute \(x = 1\), \(y = 2\), and \(z = 3\) into the equation.

Step 2: Calculate the Left Side of the Equation

Substitute \(x = 1\), \(y = 2\), and \(z = 3\) into the equation:\[3(1) + 2(2) + 3 = 3 + 4 + 3.\]Now perform the arithmetic to find the sum: \[3 + 4 + 3 = 10.\]

Step 3: Compare with the Right Side of the Equation

The result from Step 2 gives us \(10\) on the left side of the equation. The right side of the equation is already \(10\).

Step 4: Conclusion

Since the left side \(10\) equals the right side \(10\), the ordered triple \((1, 2, 3)\) does satisfy the equation \(3x + 2y + z = 10\).

Key Concepts

Linear Equations

In mathematics, linear equations form the backbone of algebraic expressions. A linear equation is any equation that, when graphed, results in a straight line. This can be understood with a simple expression such as \(ax + by + cz = d\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\), \(y\), and \(z\) are variables. The variables are only raised to the power of one, meaning there are no squares, cubes, or higher powers.
Linear equations appear in various forms but typically include two or more variables that are related by a linear function. Solving linear equations involves finding specific values for the variables that make the equation true. In the context of our exercise, we're dealing with a three-variable linear equation: \(3x + 2y + z = 10\).
Understanding and solving linear equations is fundamental as they apply to different fields, from physics to business modeling. They also serve as the foundation for more complex equations and mathematical concepts like matrices and systems of equations.

Substitution Method

The substitution method is a technique used to solve equations, especially when dealing with systems of equations. This strategy involves substituting a variable with its equivalent from another equation or an expression. It's a step-by-step approach that can simplify the process of finding variable values.
To utilize the substitution method, you first solve one of the equations for one variable. In cases of a single equation, as in our exercise, you substitute each variable with specific values, such as an ordered triple. For instance, if we have \((x, y, z) = (1, 2, 3)\), these values replace \(x\), \(y\), and \(z\) in the equation. The substitution transforms the problem into simple arithmetic.
The goal is to simplify the equation to a state where one can directly compare and confirm if both sides of the equation hold true. This makes the substitution method not only efficient but essential in verifying or solving equations where variable values are known or can be assumed.

Equation Solving

Equation solving is all about finding the values of variables that satisfy an equation. It involves several techniques, including substitution, elimination, and graphing. Problem-solving with equations allows us to find unknown values by restructuring and simplifying the given mathematical statements.
When solving an equation like \(3x + 2y + z = 10\), follow these steps:

• Identify what you need to solve - the ordered triple \((1, 2, 3)\) and its compatibility with the equation.
• Substitute these values into the equation — replace \(x\) with 1, \(y\) with 2, and \(z\) with 3.
• Simplify the equation by calculating the arithmetic result of the substitutions.

Once you reach a simplified form, compare both sides. In our solution, substituting gives us \(3 + 4 + 3 = 10\), which satisfies the original equation.
Equation solving is not just about finding a solution; it's about understanding, verifying, and manipulating mathematical statements to extract meaningful results. This is why mastery in solving equations remains crucial across various disciplines.