Problem 3 - A Graphical Approach to College Algebra

Question

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned}(-0.2,0.4,0.5)\\\&\begin{aligned} 5 x-y+2 z &=-0.4 \\\x+4 z &=1.8 \\\\-3 y+z &=-0.7\end{aligned}\end{aligned}$$

Step-by-Step Solution

Answer

Yes, \((-0.2, 0.4, 0.5)\) is a solution of the system.

Step 1: Substitute into the first equation

Take the first equation, \(5x - y + 2z = -0.4\), and substitute \(x = -0.2\), \(y = 0.4\), and \(z = 0.5\). This yields: \(5(-0.2) - 0.4 + 2(0.5)\). Calculate each term: \(-1 - 0.4 + 1 = -0.4\). This matches the right side of the equation, so the first equation is satisfied.

Step 2: Substitute into the second equation

Now take the second equation, \(x + 4z = 1.8\), and substitute \(x = -0.2\) and \(z = 0.5\). This gives: \(-0.2 + 4(0.5) = -0.2 + 2 = 1.8\). This matches the right side of the equation, so the second equation is satisfied.

Step 3: Substitute into the third equation

Finally, for the third equation, \(-3y + z = -0.7\), substitute \(y = 0.4\) and \(z = 0.5\). This gives: \(-3(0.4) + 0.5 = -1.2 + 0.5 = -0.7\). This matches the right side of the equation, so the third equation is satisfied.

Step 4: Conclusion: Verify the solution

Since the ordered triple \((-0.2, 0.4, 0.5)\) satisfies all three equations of the system, it is indeed a solution to the system of equations.

Key Concepts

Ordered Triple

An ordered triple is a simple way to express a solution for a system of three equations with three variables. The ordered triple is typically written as \((x, y, z)\), where each component represents the value of one of the variables. This notation indicates a point in a 3D space that satisfies three specific mathematical equations simultaneously. When solving systems of equations, it can sometimes be initially challenging to find these values. However, once these values are found, they offer a straightforward way to verify the solution by substitution. For instance, given the ordered triple \((-0.2, 0.4, 0.5)\), it means:

• \(x = -0.2\)
• \(y = 0.4\)
• \(z = 0.5\)

This set of values should satisfy each equation in the system when substituted back into the equations. If each equation is satisfied, we confirm that the ordered triple is indeed a valid solution.

Substitution Method

The substitution method is a handy technique for solving systems of equations, especially when equations are linear or can be manipulated into a linear form. This approach involves solving one of the equations for a particular variable and then substituting that expression into the remaining equations. Doing so reduces the number of variables and simplifies the problem. For our exercise, given the ordered triple \((-0.2, 0.4, 0.5)\), we substitute these values into each equation to check if they satisfy the system. For example:

• First Equation: Substitute values into \(5x - y + 2z = -0.4\).
• Second Equation: Use the values in \(x + 4z = 1.8\).
• Third Equation: Plug in values into \(-3y + z = -0.7\).

By substituting each value, we verify whether the initial equation holds true. This method not only helps check a solution but is also foundational in deriving solutions for systems of equations.

Algebraic Verification

Algebraic verification is the process of confirming that a proposed solution actually satisfies the given system of equations. It's like a mathematical proof, showing that each equation holds true when the given values are plugged in. This is crucial in algebra, as mistaken solutions can often be caught early through verification. For example, to verify the solution \((-0.2, 0.4, 0.5)\), substitute these values one at a time:

• Check the first equation: Confirm that substituting gives \(-0.4\), matching the right side.
• Verify the second equation: Substitution should result in \(1.8\), ensuring accuracy.
• Lastly, confirm the third equation: Results must align with the value of \(-0.7\).

Each equation must be satisfied for the solution to be correct. Using algebraic verification ensures confidence in the solution and helps prevent errors in mathematical problem-solving.