Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} -0.5 x+4 y &=7.8 \\ 0.2 x-1.6 y &=-3.6 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution for the system is x = 16.8, y = -0.15.

Step 1: Rearrange Equation 1 to Solve for x

The first equation is \( -0.5x+4y=7.8 \). If we rearrange this equation to solve for x, we have: \( x = 2(7.8 - 4y) \), or \( x = 15.6 - 8y \).

Step 2: Substitute x into Equation 2

Substituting \( x = 15.6 - 8y \) into the second equation, we get \( 0.2(15.6 - 8y) - 1.6y = -3.6 \), which simplifies to \( 3.12 - 1.6y - 1.6y = -3.6 \) or \( 3.12 - 3.2y = -3.6 \).

Step 3: Solve for y

We can now solve for y. Adding 3.2y on both sides and 3.6 on both sides, we got: \( 3.2y = -0.48 \) divinding by 3.2 we find that \( y = -0.15 \).

Step 4: Solve for x

Substitute y back into the expression for x: \( x = 15.6 - 8(-0.15) \). So \( x = 15.6 + 1.2 = 16.8 \).

Key Concepts

Substitution Method

The substitution method is a handy tool for solving systems of equations, where one equation is manipulated to express one variable in terms of the others. Here’s how it helps:

• **Rearrange an Equation**: Start by rearranging one of the equations to solve for one variable. In our exercise, we rearranged the first equation to express \( x \) in terms of \( y \).
• **Substitute the Expression**: Take the expression you found and substitute it into the other equation. This transforms a system with two variables into a single equation with one variable.
• **Solve for the Remaining Variable**: Simplify the resulting equation to find the value of the remaining variable. For instance, solving for \( y \) gave us \( y = -0.15 \).

The substitution method is especially useful for systems of equations where one equation is already or easily rearranged to express one variable in terms of the others. It reduces the number of variables in equations, simplifying the overall problem.

Graphing Utility

Using a graphing utility can be a powerful way to visually verify the solutions you've calculated using algebraic methods. Here’s how a graphing utility becomes indispensable:

• **Plotting the Equations**: Input each equation into the graphing utility to see where they intersect. The point of intersection represents the solution to the system of equations.
• **Visual Confirmation**: Graphing gives a visual confirmation of your solution. If the lines intersect at a single point, that point coordinates (\( x, y \)) are your solution.
• **Checking for Consistency**: It helps identify any errors in calculations. If the plotted lines don't intersect as expected, revisit your algebraic solution for any potential missteps.

While the substitution method provides an algebraic answer, graphing offers a visual solution, contributing another level of verification and understanding to your work.

Solving Linear Equations

Solving linear equations is a fundamental skill in mathematics, especially when dealing with systems of equations. Here's the basic process breakdown:

• **Identify Variables**: Ensure you clearly identify and define the variables involved. In the example, \( x \) and \( y \) represent the unknowns.
• **Isolating the Variable**: Use operations to isolate one variable on one side of the equation to find its value. This may involve adding, subtracting, multiplying, or dividing both sides by the same number.
• **Substituting Back**: After finding one variable's value, substitute it back into any of the original equations to find the value of the other variable. This double-checks the solution.

Being adept with solving linear equations empowers you to tackle systems of equations with ease, making you more confident in both mathematical analysis and problem-solving.