Problem 80 - Algebra 2

Question

Solve each system of equations. $$ \begin{array}{l}{4 a-3 b=-4} \\ {3 a-2 b=-4}\end{array} $$

Step-by-Step Solution

Answer

\(a = -4\), \(b = -4\)

Step 1: Choose a Variable to Eliminate

Let's eliminate the variable \(a\) from the two equations by using the method of elimination.

Step 2: Make Both Coefficients Equal

To eliminate \(a\), first, multiply the first equation by 3 and the second equation by 4 to get the coefficients of \(a\) to be the same:First equation: \(3(4a - 3b) = 3(-4)\) which simplifies to \(12a - 9b = -12\).Second equation: \(4(3a - 2b) = 4(-4)\) which simplifies to \(12a - 8b = -16\).

Step 3: Eliminate \(a\)

Subtract the second modified equation from the first modified equation:\[(12a - 9b) - (12a - 8b) = -12 - (-16)\]This simplifies to \(-b = 4\).

Step 4: Solve for \(b\)

From \(-b = 4\), multiply both sides of the equation by -1 to solve for \(b\):\[ b = -4 \]

Step 5: Substitute \(b\) into an Original Equation

Now substitute \(b = -4\) back into one of the original equations. Let's use the first one:\[ 4a - 3(-4) = -4 \]This simplifies to \(4a + 12 = -4\).

Step 6: Solve for \(a\)

Subtract 12 from both sides of the equation:\[ 4a = -16 \]Now, divide both sides by 4:\[ a = -4 \]

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations with the same set of variables. In this case, we are dealing with a system of linear equations comprised of two equations:

• \(4a - 3b = -4\)
• \(3a - 2b = -4\)

The main goal when working with systems of equations is to find the values of the variables that satisfy all equations simultaneously. There are several methods to solve these systems, including substitution, graphing, and the elimination method, which we are using here. By understanding the relationships between these equations, we can strategically manipulate them to find a solution.

Solving for Variables

Solving for variables involves finding the values for unknowns in an equation. In our system, we needed to find the values of \(a\) and \(b\) that satisfy both equations. Through strategic manipulation, we set up the equations such that variables could be eliminated.
First, we chose to eliminate \(a\) by adjusting the coefficients of \(a\) in both equations to be equal. This involved multiplying:

• The first equation by 3, changing it to \(12a - 9b = -12\)
• The second equation by 4, leading to \(12a - 8b = -16\)

With \(a\) having the same coefficient in both equations, subtraction allowed us to eliminate it and solve for \(b\). Simplifying the equation by subtracting gives us \(-b = 4\), from which we solved \(b = -4\). Understanding how operations like multiplication and subtraction affect equations is key to solving for variables effectively.

Algebraic Manipulation

Algebraic manipulation is the process of applying algebraic rules to simplify complex equations by using operations such as addition, subtraction, multiplication, and division. Here, algebraic manipulation played a critical role in solving the system of equations.
First, we aligned the equations by making sure they had the same coefficient with respect to the selected variable (\(a\) in this case). We did this through multiplication:

• First equation: Multiplied by 3 -> \(12a - 9b = -12\)
• Second equation: Multiplied by 4 -> \(12a - 8b = -16\)

Then, we subtracted the second equation from the first to remove \(a\), simplifying to \(-b = 4\), allowing us to solve for \(b\).
Upon finding \(b\), we substituted back into one of the original equations to solve for \(a\). This is a classic demonstration of how algebraic manipulation is used to isolate and solve for variables, an essential skill in algebra.