Problem 40 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}\frac{1}{4} s-\frac{2}{3} t=-3 \\ \frac{1}{3} s+\frac{1}{3} t=7\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \( (s, t) = (12, 9) \).

Step 1: Solve One Equation for a Variable

We'll solve the second equation for \( s \) in terms of \( t \). The second equation is:\[ \frac{1}{3}s + \frac{1}{3}t = 7 \]Subtract \( \frac{1}{3}t \) from both sides:\[ \frac{1}{3}s = 7 - \frac{1}{3}t \]Multiply every term by 3 to cancel the fraction:\[ s = 21 - t \]

Step 2: Substitute the Expression into the Other Equation

Substitute \( s = 21 - t \) into the first equation:\[ \frac{1}{4}(21 - t) - \frac{2}{3}t = -3 \]Simplify the substitution:\[ \frac{21}{4} - \frac{1}{4}t - \frac{2}{3}t = -3 \]

Step 3: Clear Fractions by Finding a Common Denominator

The common denominator for 4 and 3 is 12. Multiply the entire equation by 12 to clear the fractions:\[ 12 \times \left( \frac{21}{4} - \frac{1}{4}t - \frac{2}{3}t \right) = 12 \times (-3) \]Which simplifies to:\[ 63 - 3t - 8t = -36 \]

Step 4: Combine Like Terms

Combine the \( t \) terms:\[ 63 - 11t = -36 \]

Step 5: Isolate the Variable

Subtract 63 from both sides to solve for \( t \):\[ -11t = -36 - 63 \]Simplifying this gives:\[ -11t = -99 \]Divide both sides by -11:\[ t = 9 \]

Step 6: Substitute Back to Find Other Variable

Substitute \( t = 9 \) back into the equation \( s = 21 - t \):\[ s = 21 - 9 \]Simplify to get:\[ s = 12 \]

Step 7: Verify the Solution

Substitute \( s = 12 \) and \( t = 9 \) back into the original equations to verify the solution.For the first equation:\[ \frac{1}{4}(12) - \frac{2}{3}(9) = 3 - 6 = -3 \] which is correct.For the second equation:\[ \frac{1}{3}(12) + \frac{1}{3}(9) = 4 + 3 = 7 \] which is also correct.Thus the solution \( (s, t) = (12, 9) \) satisfies both equations.

Key Concepts

Solving Systems of Equations

Solving systems of equations involves finding values for variables that satisfy each equation simultaneously. It's like finding a common solution to multiple mathematical puzzles at once. In the substitution method, one of the equations is solved for one variable in terms of the other variable. This expression is then substituted into the other equation, transforming a system of equations into a single one. Here's an easy way to approach this:

• Choose the simpler equation for substitution.
• Solve it for one variable to express it in terms of the other.
• Substitute this result into the other equation.

By doing so, you can solve the system step by step, eventually finding the values for both variables. This process helps simplify complex systems, allowing us to focus on one relationship at a time.

Fractions in Algebra

Fractions in algebra can be intimidating, but knowing how to handle them can simplify solving equations greatly. When you encounter fractions, finding a common denominator can make subtraction or addition easier. However, in equations, multiplying through by a least common denominator is often the best strategy to eliminate fractions altogether.

• Identify denominators in your equation.
• Determine the least common denominator (LCD) that encompasses all fractions.
• Multiply every term in the equation by this LCD.

This step transforms your fractions into whole numbers, making the remaining algebraic steps more straightforward. It’s essential to practice this technique to foster fluency with fractions and bolster your overall problem-solving skills in algebra.

Step-by-Step Algebra Solutions

Step-by-step algebra solutions guide you through solving a problem without feeling overwhelmed. Each step breaks down the problem into manageable parts, helping you focus on one aspect at a time. Here’s how you can tackle an equation system with clarity and confidence.

• Simplify first: Start by simplifying expressions to remove fractions or reduce terms.
• Use substitution wisely: Solve one equation for one variable whenever possible.
• Isolate variables: Focus on finding a single variable first, then use it to solve for others.
• Verify your solution: Always plug your solution back into the original equations to ensure they satisfy all conditions.

Following each of these steps builds a comprehensive understanding of the problem and ensures you don’t miss critical details. Practicing this structured approach will enhance your capability to solve more complex algebraic problems effectively.