Problem 36 - Algebra 1

Question

Use linear combinations to solve the system of linear equations. $$\begin{array}{l} 3 a+9 b=8 b-a \\ 5 a-10 b=4 a-9 b+5 \end{array}$$

Step-by-Step Solution

Answer

The solutions to the system of equations are \(a = 1\) and \(b = -4\).

Step 1: Simplify the equations

First, simplify each of the given equations: For Equation 1: \(3a + 9b = 8b - a\) This simplifies to: \(4a + b = 0\).For Equation 2: \(5a - 10b = 4a - 9b + 5\) This simplifies to: \(a - b = 5\).

Step 2: Solve for a

From the simplified form of Equation 2: \(a - b = 5\), we can solve for a by rearranging the equation \(a = b + 5\)

Step 3: Substitute a in Equation 1

Substitute \(a = b + 5\) into the simplified form of Equation 1: \(4a + b = 0\) to get: \(4(b + 5) + b = 0\). This simplifies to \(5b = -20\)

Step 4: Solve for b

Solving \(5b = -20\), we get \(b = -4\)

Step 5: Substitute b into Equation 2

Finally, substitute \(b = -4\) into Equation 2 to solve for a: \(a = -4 + 5 = 1\)

Key Concepts

Linear Combinations

Linear combinations are a method used to solve systems of linear equations, by adding or subtracting equations to eliminate one or more of the unknowns. In a system of equations, we aim to make coefficients of one variable the same in two equations, so they can easily cancel out. This allows us to solve for the remaining unknown.

• Addition or Subtraction: We add or subtract either whole equations or parts of them, to reduce the number of unknowns.
• Elimination: The goal is to eliminate one variable at a time until only one variable is left.

This technique is potent when working with two or more equations that share variables. Each linear equation represents a line, and solving the system finds their intersection point, where both conditions (equations) are satisfied. In the given problem, we use linear combinations to turn our given equations into something simpler before solving.

System of Equations

A system of equations consists of several equations that share the same set of variables. Here, we are dealing with a system of two linear equations:

• Equation 1: \(4a + b = 0\)
• Equation 2: \(a - b = 5\)

The solution to a system of equations is the values for each variable that make all equations true. Systems of equations can be solved using various methods, such as substitution, elimination (linear combination), or graphing. For our task, we found solutions to both equations that intersect at a point (a solution pair \((a, b)\)). Understanding systems of equations is crucial as they represent multiple constraints within a single problem. This concept is widely applied in fields like economics, engineering, and science for modeling real-world situations.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. Once simplified, equations become more manageable to solve by applying basic algebraic operations.
To solve the system, we:

• Isolated variables: We rearranged Equation 2 to express \(a\) in terms of \(b\): \(a = b + 5\).
• Substituted: Substituted \(a = b + 5\) into Equation 1 and solved for \(b\) to find \(b = -4\).
• Back-substitution: Used the value of \(b\) to find \(a = 1\).

These steps highlight the systematic approach of solving any equation or set of equations by transforming them into simpler forms. Being comfortable with these methodologies helps in tackling more complex algebraic problems.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations using basic algebraic operations such as addition, subtraction, multiplication, and division to simplify them or make them easier to solve.

• Simplify: Combine like terms and perform operations on both sides of the equations to reduce complexity.
• Rearrange: Move terms around to isolate a variable on one side of the equation.
• Factor: Sometimes it helps to factorize expressions to simplify further solving steps or to find roots.

In the given problem, algebraic manipulation allowed us to take a system of potentially complex equations and reduce them down into simpler, solvable forms. Proper understanding and use of these skills are foundational for anyone seeking to master algebra and solve linear equations effectively in any context.