Problem 50 - Algebra 2

Question

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{3 x+8 y=23} \\ {x-y=4}\end{array} $$

Step-by-Step Solution

Answer

The solution is \((5, 1)\).

Step 1: Write down the system of equations

We begin with the given system of equations: 1. \( 3x + 8y = 23 \) 2. \( x - y = 4 \)

Step 2: Solve one equation for one variable

Choose the second equation to isolate one of the variables. We will solve for \( x \) in terms of \( y \):\[ x = y + 4 \]

Step 3: Substitute the expression into the other equation

Substitute \( x = y + 4 \) into the first equation (\( 3x + 8y = 23 \)):\[ 3(y + 4) + 8y = 23 \]

Step 4: Simplify and solve for y

Distribute and combine like terms:\[ 3y + 12 + 8y = 23 \]\[ 11y + 12 = 23 \]Subtract 12 from both sides:\[ 11y = 11 \]Divide by 11:\[ y = 1 \]

Step 5: Substitute the value of y back into the expression for x

Now substitute \( y = 1 \) back into the expression for \( x \):\[ x = y + 4 \]\[ x = 1 + 4 \]\[ x = 5 \]

Step 6: Write the solution to the system of equations

The solution to the system of equations is \( x = 5 \) and \( y = 1 \). The ordered pair is \((5, 1)\).

Key Concepts

substitution method

The substitution method is a technique for solving systems of linear equations. It involves first solving one of the equations for one variable. After isolating that variable, you then substitute this expression into the other equation. This reduces the number of variables in the equation, allowing for simplification and solution.
Here's how you can think of it:

• Solve one equation for one variable. Simplifying one equation first makes the next steps easier.
• Substitute this expression into the other equation. This narrows the focus of the problem.
• Solve the simplified equation. You can resolve it to find the value of the remaining variable.
• Finally, substitute back to find the unknown value of the first variable.

This technique is beneficial when one of the equations is easily solvable for a particular variable. It's a straightforward approach that breaks down complex systems into simpler, sequential steps, making problem-solving more manageable.

elimination method

The elimination method is another effective strategy for solving systems of linear equations. This method aims to eliminate one of the variables by adding or subtracting the equations. The goal is to create a new equation with just one variable, simplifying the system.
Here's a simplified approach:

• Align the equations so the variables you want to eliminate are in the same position.
• Add or subtract the equations to remove one of the variables.
• Solve the new single-variable equation. This gives the value of one variable.
• Substitute the found variable back into one of the original equations to solve for the other variable.

This method is particularly useful when coefficients of one of the variables are easy to adjust so they result in complete elimination. This systematic approach simplifies complex calculations and enhances precision in finding solutions.

solving linear equations

Solving linear equations is a fundamental skill in mathematics, often used to find the values of variables that satisfy conditions given by one or more equations. Linear equations are equations where variables appear only in the power of one, meaning there are no squared terms, cube roots, or other complex operations.
The steps to solve basic linear equations generally include:

• Isolating the variable: Get all terms with the desired variable on one side of the equation.
• Simplification: Use operations like addition, subtraction, multiplication, and division to simplify the equation.
• Checking the solution: Substitute the solution back into the original equation to ensure it satisfies the equation.

These steps are applicable whether solving a single equation or part of a larger system. Mastering this process improves overall problem-solving abilities in various mathematical contexts.

mathematics problem solving

Mathematics problem solving is the process of finding a solution to a challenging question that requires strategic use of various mathematical concepts and methods. It involves understanding the problem, planning a strategy, executing the solution, and reflecting on the solution's validity.
Here’s a guided plan for mathematics problem solving:

• Understand the problem: Read it carefully and determine what is being asked.
• Devise a plan: Decide which strategies or methods you can use to solve the problem.
• Carry out the plan: Carefully implement your chosen strategies and solve the problem.
• Review/Reflect: Check if your solution solves the problem correctly, and think about other ways the problem could be approached.

This systematic approach ensures you remain organized and efficient in tackling a wide variety of mathematical problems.