Problem 88 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{l} 2 x-3 y=1 \\ 3 x-2 y=2 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{5}\) and \(y = \frac{1}{5}\).

Step 1: Write the System of Equations

We have two equations in the system, which are: 1. \(2x - 3y = 1\) 2. \(3x - 2y = 2\). The goal is to find values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Use the Elimination Method

To eliminate one variable, we'll multiply the first equation by 3 and the second equation by 2, making the coefficient of \(x\) the same in both equations. Multiply the first equation by 3:\[3(2x - 3y) = 3(1)\]\[6x - 9y = 3\]Multiply the second equation by 2:\[2(3x - 2y) = 2(2)\]\[6x - 4y = 4\].

Step 3: Subtract the Equations

Subtract the second transformed equation from the first transformed equation to eliminate \(x\):\[(6x - 9y) - (6x - 4y) = 3 - 4\]\[-9y + 4y = -1\]\[-5y = -1\].

Step 4: Solve for y

Now solve the equation \(-5y = -1\) for \(y\):Divide both sides by -5:\[y = \frac{1}{5}\].

Step 5: Substitute y and Solve for x

Substitute \(y = \frac{1}{5}\) into the first equation to find \(x\):\[2x - 3\left(\frac{1}{5}\right) = 1\]\[2x - \frac{3}{5} = 1\].Add \(\frac{3}{5}\) to both sides:\[2x = 1 + \frac{3}{5}\]\[2x = \frac{8}{5}\].Divide both sides by 2:\[x = \frac{8}{5} \times \frac{1}{2}\]\[x = \frac{4}{5}\].

Step 6: Verify the Solution

Substitute \(x = \frac{4}{5}\) and \(y = \frac{1}{5}\) into the second equation to verify:\[3\left(\frac{4}{5}\right) - 2\left(\frac{1}{5}\right) = 2\]\[\frac{12}{5} - \frac{2}{5} = 2\]\[\frac{10}{5} = 2\]\[2 = 2\].The solution satisfies both equations.

Key Concepts

Elimination Method

The elimination method is a popular algebraic technique to solve systems of linear equations. It involves manipulating equations to cancel out one of the variables, making it easier to solve for the remaining one.

• Start by aligning the variables of the two equations vertically, ensuring related terms like those with variable x are directly above each other.
• Multiply one or both equations by numbers that will make the coefficient of one of the variables the same in both equations. For instance, if you have two equations like \(2x - 3y = 1\) and \(3x - 2y = 2\), you can multiply them such that the coefficients of \(x\) become equal.
• Once the coefficients are matched, subtract or add the equations to eliminate that variable, leaving only one variable in the equation.
• Next, solve the resulting equation for the remaining variable.

This method is particularly useful because it simplifies the problem, reducing two variables to just one at a time. It works well with both simple and complex systems and ensures clarity in the solution process.

Algebraic Solution

An algebraic solution refers to using algebraic operations and methods to find the values of unknown variables. In the context of solving two linear equations, algebraic solutions often involve rearranging and combining equations to find the answer.

• After eliminating one variable using the elimination method, solve for the remaining variable using basic algebraic techniques such as addition, subtraction, multiplication, or division.
• Once you find the value of one variable, substitute it back into one of the original equations to find the other variable.
• Ensure you handle fractions and decimals with care to maintain accuracy throughout the calculations.

The algebraic solution process is systematic and logical, helping you track each step and verify the solutions for consistency with the original equations. It builds a strong foundation for tackling more complicated algebraic problems in future studies.

Linear Equations

Linear equations present a foundation for understanding algebra and its application in solving real-world problems. These equations are called "linear" because they represent straight lines when graphed on a coordinate plane.

• Each linear equation in two variables represents a line where each solution of the equation corresponds to a point on that line.
• A linear equation typically takes the form \(ax + by = c\), where \(x\) and \(y\) are variables, and \(a\), \(b\), and \(c\) are constants. The solution to a system of linear equations is where the lines intersect.
• Understanding how to manipulate and solve these equations is critical as it allows for finding where two conditions, represented by the equations, meet.

Grasping linear equations helps in visualizing the problems as geometric figures, providing a tangible way of interpreting and solving mathematical problems. Mastering this concept is crucial for advanced topics in mathematics and related disciplines.