Problem 7 - Intermediate Algebra

Question

If the system\(\left\\{\begin{array}{l}4 x-3 y=7 \\ 3 x-2 y=6\end{array}\right.\)by what number should each equation be multiplied if a. the \(x\) -terms are to drop out? b. the \(y\) -terms are to drop out?

Step-by-Step Solution

Answer

a. Multiply first equation by 3, second by 4. b. Multiply first equation by 2, second by 3.

Step 1: Identify the coefficients for x

The coefficients of \(x\) in the given system are \(4\) and \(3\). To eliminate \(x\) by subtraction, make the \(x\) coefficients the same. This requires finding a common multiple.

Step 2: Find common multiple for x

The least common multiple (LCM) of \(4\) and \(3\) is \(12\). Each equation should be multiplied by a number that will make the \(x\) terms equal to \(12\).

Step 3: Determine multiplication factors for x

Multiply the first equation by \(3\) (since \(4 \times 3 = 12\)) and the second equation by \(4\) (since \(3 \times 4 = 12\)). This will eliminate the \(x\) terms when the equations are subtracted from one another.

Step 4: Identify the coefficients for y

The coefficients of \(y\) in the given system are \(-3\) and \(-2\). To eliminate \(y\), find the least common multiple of their absolute values.

Step 5: Find common multiple for y

The least common multiple (LCM) of \(3\) and \(2\) is \(6\). Each equation should be multiplied by a number that will make the \(y\) terms equal to either \(6\) or \(-6\).

Step 6: Determine multiplication factors for y

Multiply the first equation by \(2\) (since \(-3 \times 2 = -6\)) and the second equation by \(3\) (since \(-2 \times 3 = -6\)) to eliminate the \(y\) terms when the equations are subtracted from one another.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations, where the goal is to eliminate one of the variables by combining the equations. This method involves adjusting the equations so that adding or subtracting them results in one variable disappearing, allowing you to solve for the remaining variable. It is often preferred due to its systematic approach.

In the given exercise, we aim to eliminate either the \(x\) or \(y\) variable to solve the system:

• \(4x - 3y = 7\)
• \(3x - 2y = 6\)

To achieve this, each equation is multiplied by appropriate factors so that one of the variables will have equal and opposite coefficients, enabling it to be canceled out through subtraction. This leaves us with a simpler equation to solve. After finding the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable.

Least Common Multiple

The least common multiple (LCM) is a concept used in mathematics to find the smallest multiple that two numbers have in common. When working with systems of linear equations, the LCM helps in aligning the coefficients of a chosen variable across different equations so that it can be eliminated.

In our exercise, the least common multiple plays a crucial role:

• For the \(x\)-terms with coefficients 4 and 3, the LCM is 12.
• For the \(y\)-terms with coefficients -3 and -2, the LCM is 6.

By identifying these LCMs, the equations can be multiplied by specific factors to equalize the coefficients of either \(x\) or \(y\) in both equations. This sets the stage for using subtraction to eliminate a variable, simplifying the solving process.

Linear Combinations

Linear combinations in the context of solving equations involve creating new equations by adding or subtracting multiples of existing equations. This technique is useful in manipulating a system of equations so that one variable can be easily eliminated.In this task, we form a linear combination to eliminate a variable:

• For \(x\): Multiply the first equation by 3 (\(4 \times 3 = 12\)) and the second by 4 (\(3 \times 4 = 12\)). By aligning their \(x\) coefficients, subtraction eliminates \(x\).
• For \(y\): Multiply the first equation by 2 (\(-3 \times 2 = -6\)) and the second by 3 (\(-2 \times 3 = -6\)). Aligning their \(y\) coefficients similarly allows their elimination.

This process of forming linear combinations through multiplication and subtraction or addition is fundamental to simplifying and solving systems of linear equations efficiently.