Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Matrix Multiplication Use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. \(\left\\{\begin{aligned}-2 x-3 y &=-6 \\ 4 x+2 y &=20 \end{aligned}\right.\) (a) \(\left[\begin{array}{l}3 \\ 0\end{array}\right]\) (b) \(\left[\begin{array}{l}4 \\ 2\end{array}\right]\) (c) \(\left[\begin{array}{r}-6 \\ 6\end{array}\right]\) (d) \(\left[\begin{array}{r}6 \\ -2\end{array}\right]\)

Step-by-Step Solution

Answer

Only the solution (d) \([6 \\ -2]\) satisfies the system of equations.

Step 1: Test the case (a) \([3 \\ 0]\)

Substitute \(x = 3\) and \(y = 0\) into each equation. After substitution, the system looks like: \[\{-2*3 - 3*0 = -6\\4*3 + 2*0 = 20\}\] Simplifying gives us: \[\{-6 = -6\\12 = 20\}\] Clearly, the second equation doesn't hold, so solution (a) doesn't satisfy the system.

Step 2: Test the case (b) \([4 \\ 2]\)

Substitute \(x = 4\) and \(y = 2\) into each equation. After substitution, the system looks like: \[\{-2*4 - 3*2 = -6\\4*4 + 2*2 = 20\}\] Simplifying gives us: \[\{-14 = -6\\20 = 20\}\] The first equation doesn't hold, so solution (b) is not a solution for the system.

Step 3: Test the case (c) \([-6 \\ 6]\)

Substitute \(x = -6\) and \(y = 6\) into each equation. After substitution, the system looks like: \[\{-2*(-6) - 3*6 = -6\\4*(-6) + 2*6 = 20\}\] Simplifying gives us: \[0 = -6\\0 = 20\] Neither equation holds, so solution (c) doesn't satisfy the system.

Step 4: Test the case (d) \([6 \\ -2]\)

Substitute \(x = 6\) and \(y = -2\) into each equation. After substitution, the system looks like: \[\{-2*6 - 3*(-2) = -6\\4*6 + 2*(-2) = 20\}\] Simplifying gives us: \[-6 = -6\\20 = 20\] Both equations hold, so solution (d) is a valid solution for the system.

Key Concepts

Systems of Equations

Systems of equations are like puzzles that need solving to find values that work for all the equations in the system. Each equation represents a straight line, and the solution to the system is the point or points where these lines intersect. In our example, we have a system of two equations:

• \(-2x - 3y = -6\)
• \(4x + 2y = 20\)

We need to find values of \(x\) and \(y\) that make both equations true at the same time. In other words, we’re looking for the intersection of two lines on a graph. The solution will be a pair of numbers, one for \(x\) and one for \(y\). Understanding this can help you visualize what the equations mean and find the answer more easily.

Graphing Utility Verification

Using a graphing utility can be a helpful way to verify solutions to systems of equations. A graphing utility can be a calculator or computer program that will plot the lines represented by the equations. Once the lines are graphed, you simply look for their point of intersection.
In our case:

• The first line, \(-2x - 3y = -6\), is graphed.
• Then, the second line, \(4x + 2y = 20\), is plotted.

The intersection point of these lines will reveal the solution to our system. This visual method allows for an easy check of the algebraic solution by verifying if the test points lie on the intersection.

Substitution Method

The substitution method is solving systems of equations by finding an expression for one variable and substituting it into the other equation. It’s like breaking down a problem into simpler steps. Start with the equation that’s easiest to solve for one of the variables. Substitute this expression into the other equation.
For our problem, we tested substitution directly by plugging in different values of \(x\) and \(y\):

• Test (a) with \(x = 3, y = 0\)
• Test (b) with \(x = 4, y = 2\)
• Test (c) with \(x = -6, y = 6\)
• Test (d) with \(x = 6, y = -2\)

When we substitute these values, we check if both equations in the system are true. Only when both hold true, those values are the solution of the system.

Equation Solutions

Finding the solution to an equation involves some simple arithmetic. When you substitute values into the system of equations, simplify both equations:1. Multiply and simplify each equation.2. Check if the simplified expression equals the constant on the other side of the equation.

• If it matches for both equations, it’s a solution.
• If even one doesn't match, that point isn't a solution.

In our steps, we found each pair of \((x, y)\) until we discovered that pair \((6, -2)\) satisfies both equations. This means it is the correct solution to the system, where the values work flawlessly in both equations.

• For \(\[ -2*6 - 3*(-2) = -6 \] \) it checks out.
• For \(\[ 4*6 + 2*(-2) = 20 \] \) it does as well.

Finding these matches confirms the solution is correct.