Problem 44 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{rl}\frac{1}{2} x+\frac{2}{3} y & =-\frac{3}{10} \\ 5 x+4 y & =-1\end{array}\right)\)

Step-by-Step Solution

Answer

The solution to the system is \(x = \frac{2}{5}\), \(y = -\frac{3}{4}\).

Step 1: Solve First Equation for x

Start by solving the first equation for one of the variables. Let's solve for \(x\):\[\frac{1}{2}x + \frac{2}{3}y = -\frac{3}{10}\]Subtract \(\frac{2}{3}y\) from both sides:\[\frac{1}{2}x = -\frac{3}{10} - \frac{2}{3}y\]Multiply both sides by 2 to solve for \(x\):\[x = -\frac{6}{10} - \frac{4}{3}y\]

Step 2: Substitute x in Second Equation

Substitute the expression for \(x\) from the first step into the second equation:\[5\left(-\frac{6}{10} - \frac{4}{3}y\right) + 4y = -1\]Simplify:\[-\frac{30}{10} - \frac{20}{3}y + 4y = -1\]Simplify further:\[-3 - \frac{20}{3}y + \frac{12}{3}y = -1\]Combine the \(y\) terms:\[-3 - \frac{8}{3}y = -1\]

Step 3: Solve for y

Now, solve the equation from Step 2 for \(y\):\[-3 - \frac{8}{3}y = -1\]Add 3 to both sides:\[-\frac{8}{3}y = 2\]Multiply both sides by \(-\frac{3}{8}\) to solve for \(y\):\[y = -\frac{3}{4}\]

Step 4: Substitute y Back to Find x

Now that we have \(y\), substitute \(y = -\frac{3}{4}\) back into the expression for \(x\) from Step 1:\[x = -\frac{6}{10} - \frac{4}{3} \left(-\frac{3}{4}\right)\]Simplify:\[x = -\frac{3}{5} + 1\]Thus, \(x = \frac{2}{5}\).

Step 5: Verify the Solution

Substitute \(x = \frac{2}{5}\) and \(y = -\frac{3}{4}\) back into both original equations to verify the solution.**Verification in the first equation:**\[\frac{1}{2}\left(\frac{2}{5}\right) + \frac{2}{3}\left(-\frac{3}{4}\right) = -\frac{3}{10}\]Calculate each term and confirm equality.**Verification in the second equation:**\[5\left(\frac{2}{5}\right) + 4\left(-\frac{3}{4}\right) = -1\]Calculate each term and confirm equality if both solutions satisfy the equations.

Key Concepts

Understanding Systems of Equations

Systems of equations can seem complex, but they simply involve finding values for variables that satisfy multiple equations at once. Each equation in the system provides a piece of information about the same variables. When we have a system of linear equations, such as the one given:

• \( \frac{1}{2} x + \frac{2}{3} y = -\frac{3}{10} \)
• \( 5x + 4y = -1 \)

it means we're looking for pairs of \( (x, y) \) that make both equations true simultaneously.
In simpler terms, imagine trying to find the point where two lines cross each other on a graph. That's what solving a system of equations entails. It's essential to choose a method that suits the problem at hand, and here, we're focusing on the substitution method. This method involves solving one of the equations for a single variable and using that expression in the second equation to find the other variable.

Algebraic Solution Through Substitution

The substitution method is a powerful tool for solving systems of equations. It involves a simple, yet precise series of steps. Let's dive into it using our example:

- **Step 1: Solve one equation for one variable.**
We chose the first equation to solve for \(x\):\[ \frac{1}{2} x + \frac{2}{3} y = -\frac{3}{10} \]Rearranging gives us \( x = -\frac{6}{10} - \frac{4}{3}y \). This provides an expression for \(x\) that we can use in the second equation.

- **Step 2: Substitute the expression into the other equation.**
Place the expression for \(x\) into the second equation:\[ 5(-\frac{6}{10} - \frac{4}{3}y) + 4y = -1 \]Continue simplifying until you isolate \(y\).

- **Step 3: Solve for the remaining variable.**
The above substitution leads to finding \( y = -\frac{3}{4} \). This value of \(y\) is crucial, as it helps us find \(x\).

Finally, insert \(y\) back into the expression we have for \(x\) to find \( x = \frac{2}{5} \). These steps demonstrate how substitution transforms solving systems into manageable tasks of algebraic manipulation.

Verification of Solutions

Verification is a logical, yet often overlooked step in solving equations. It ensures that the solutions found truly satisfy the original equations. Here's how we can check the solutions we've just found:

- **Insert the solutions back into both original equations.**

• For the first equation: Substitute \( x = \frac{2}{5} \) and \( y = -\frac{3}{4} \) into \[ \frac{1}{2}\left(\frac{2}{5}\right) + \frac{2}{3}\left(-\frac{3}{4}\right) = -\frac{3}{10} \]
Calculate each term to confirm if both sides of the equation are equal.

• For the second equation: Do the same with the expression \[ 5\left(\frac{2}{5}\right) + 4\left(-\frac{3}{4}\right) = -1 \]
Again, compute each side to verify equality.

This step is crucial for catching any mistakes made during calculation. By ensuring consistency across both equations, you confirm that \(x\) and \(y\) are indeed the intersection point where both lines coincide. It's like checking your work with a magnifying glass before submitting it!