Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} 1.5 x+0.8 y=2.3 \\ 0.3 x-0.2 y=0.1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1\) and \(y = 1\).

Step 1: Solve the Second Equation for x

To begin, reorder the second equation and solve for \(x\): \n\n\[0.3x = 0.1 + 0.2y\]\n\nDividing each side by 0.3 gives: \n\n\[x = \frac{1}{3} + \frac{2}{3}y\]

Step 2: Substitute into the First Equation

Next, substitute the expression for \(x\) from Step 1 into the first equation. This will give an equation in terms of \(y\) only.\n\n\[1.5\left(\frac{1}{3} + \frac{2}{3}y\right) + 0.8y = 2.3\]\n\nSimplifying this gives: \n\n\[0.5 + y + 0.8y = 2.3\]\n\nThis simplifies further to: \n\n\[1.8y = 1.8\]

Step 3: Solve for y

We can solve this equation for \(y\) by dividing each side by 1.8: \n\n\[y = 1\]

Step 4: Substitute y = 1 into the equation for x

Now we can substitute \(y = 1\) into the equation for \(x\) from Step 1: \n\n\[x = \frac{1}{3} + \frac{2}{3}\times1\]\n\nWhich simplifies to: \n\n\[x = 1\]

Step 5: Check the Solution

Lastly, we must verify our results. Plug \(x = 1\) and \(y = 1\) back into the original equations to check if the left side equals the right side. Upon verification, it's concluded that the results are correct.

Key Concepts

Substitution Method

The substitution method is a straightforward way to solve a system of equations. It involves expressing one variable in terms of another and substituting this expression into the other equation. This eliminates one of the variables and allows you to solve for the remaining variable.

In our given system of equations, we start by solving the second equation for the variable \(x\):

• Rearrange the second equation: \(0.3x = 0.1 + 0.2y\)
• Divide by 0.3 to find \(x\): \(x = \frac{1}{3} + \frac{2}{3}y\)

Now you have an expression for \(x\) in terms of \(y\). Substitute this expression for \(x\) into the first equation.

• Initial equation: \(1.5x + 0.8y = 2.3\)
• Substitute \(x\): \(1.5(\frac{1}{3} + \frac{2}{3}y) + 0.8y = 2.3\)

Now, the equation only contains the variable \(y\), and you can solve it directly.

Solving Equations

Solving equations is a crucial skill in mathematics, involving finding the value of the variable that makes an equation true. In the substitution method, we end up with a simpler equation with only one variable.

Continuing with our substituted equation:

• Simplify \(1.5(\frac{1}{3} + \frac{2}{3}y) + 0.8y = 2.3\)
• This becomes \(0.5 + y + 0.8y = 2.3\)
• Combine like terms: \(1.8y = 1.8\)

Now, solve for \(y\) by dividing both sides by 1.8:

• \(y = 1\)

Once we have \(y\), substitute it back into the expression for \(x\), giving us:

• \(x = \frac{1}{3} + \frac{2}{3} \times 1\)
• \(x = 1\)

Graphing Utility

A graphing utility is a tool, either software or a calculator, that helps visualize equations and their solutions. It's useful for verifying solutions because it shows where the graphs of the individual equations intersect.

To verify our solution using a graphing utility, enter both equations in the utility:

• First equation: \(1.5x + 0.8y = 2.3\)
• Second equation: \(0.3x - 0.2y = 0.1\)

Upon graphing, look for the intersection point of the two lines. This point should match the solution found algebraically: \( (x, y) = (1, 1) \). If it does, it gives visual confirmation that the solution is correct.

Verifying Solutions

Verifying solutions is the process of checking whether your solution is correct by substituting the values back into the original equations. It ensures that the solutions satisfy both equations in the system.

For our case, substitute \(x = 1\) and \(y = 1\) back into the original equations:

• First equation: \(1.5 \cdot 1 + 0.8 \cdot 1 = 2.3\)
• Second equation: \(0.3 \cdot 1 - 0.2 \cdot 1 = 0.1\)

Both equations should evaluate to true:

• Checking the first gives \(1.5 + 0.8 = 2.3\)
• Checking the second gives \(0.3 - 0.2 = 0.1\)

Seeing that both are true confirms that \((1, 1)\) is indeed the solution to the system of equations.