Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} \frac{1}{2} x+\frac{3}{4} y=10 \\ \frac{3}{4} x-y=4 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 16 and y = 8.

Step 1: Isolate One Variable

First, isolate one variable in one of the two equations. It's easier to isolate 'y' in the second equation. The second equation can be rewritten as \(y = \frac{3}{4}x - 4\).

Step 2: Substitute the Value

Now, substitute \(y = \frac{3}{4}x - 4\) into the first equation to find the value of 'x'. This gives us \(\frac{1}{2}x + \frac{3}{4}\left(\frac{3}{4}x - 4\right) = 10\)

Step 3: Solve for 'x'

Simplify and solve the above equation for 'x'. To make it easier, first find common denominator to sum terms and simplify: \(0.5x + 0.5625x - 3 = 10\), which simplifies to \(x = 16\)

Step 4: Solve for 'y'

Substitute \(x = 16\) back into the equation for y which gives \(y = \frac{3}{4} * 16 - 4 = 8\)

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of linear equations. The idea is to solve one of the equations for one variable and then substitute that expression into the other equation. This method allows us to reduce the system to a single variable, making it easier to solve.

Let's consider the given system of equations:

• \( \frac{1}{2} x + \frac{3}{4} y = 10 \)
• \( \frac{3}{4} x - y = 4 \)

In step 1, you isolate one variable. It is easier to express 'y' in terms of 'x' from the second equation: \( y = \frac{3}{4} x - 4 \).

In step 2, after isolating 'y', substitute \( y = \frac{3}{4} x - 4 \) into the first equation. This transformation changes the system into one equation that only contains 'x'. Solving this single-variable equation will provide the value of 'x'.
Using substitution helps in several ways:

• It's systematic and straightforward, leading to fewer mistakes.
• It works well when one equation can be easily solved for one variable.
• Makes it easy to handle different types of coefficients in equations.

Solving Linear Equations

Once you've substituted one variable with an expression involving the other, you have an equation with just one unknown. Solving a linear equation involves finding the variable's value that makes the equation true. For this exercise, after substituting \( y = \frac{3}{4} x - 4 \) into the first equation, you get \( \frac{1}{2}x + \frac{3}{4}\left(\frac{3}{4}x - 4\right) = 10 \).

To solve, first expand and simplify the equation:

• \( 0.5x + 0.5625x - 3 = 10 \)
• Combine like terms: \( 1.0625x - 3 = 10 \)
• Add 3 to both sides: \( 1.0625x = 13 \)
• Finally, solve for 'x' by dividing both sides by 1.0625: \( x = 16 \)

Once 'x' is found, plug it back into the expression for 'y'. With \( x = 16 \), the value of 'y' is found as follows:

• \( y = \frac{3}{4} \times 16 - 4 = 8 \)

Linear equations like these form the cornerstone of many fields in applied mathematics and science, providing essential solutions to various real-world problems.

Graphing Utility Verification

After solving a system of equations algebraically, it's beneficial to verify the solution using a graphing utility. Verification assures that the mathematical solution matches the graphical intersection point of the equations, providing confirmation and confidence in the result.

When using a graphing utility, simply input each of the given equations:

• \( \frac{1}{2}x + \frac{3}{4}y = 10 \)
• \( \frac{3}{4}x - y = 4 \)

Plot the graphs for both equations. The solution to the system is where the two lines intersect. If done accurately, the intersection should appear at the point \( (16, 8) \), confirming our algebraic solution.

Utilizing graphing tools assists students in:

• Visually understanding the nature of linear systems and solutions.
• Gaining insight into why certain solutions are valid.
• Building trust in their algebraic problem-solving skills.

By bridging algebra with visual graph-based methods, students can deepen their comprehension of mathematical relationships and verify their calculations effectively.