Problem 20 - Intermediate Algebra

Question

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{r} 4 x-7 y=21 \\ -4 x+3 y=-9 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \(x = 0\) and \(y = -3\).

Step 1: Identify the Suitable Method

For this system of equations, notice that the coefficients of \(x\) in both equations are \(4\) and \(-4\), respectively. These coefficients suggest that using the elimination-by-addition method could immediately eliminate \(x\) when the equations are added together.

Step 2: Add the Equations

Add the two given equations to eliminate \(x\):\[ (4x - 7y) + (-4x + 3y) = 21 + (-9) \]. The \(4x\) and \(-4x\) terms will cancel each other out.

Step 3: Simplify the Resulting Equation

After adding the equations, simplify the resulting expression: \( -4y = 12 \).

Step 4: Solve for y

Solve the equation \(-4y = 12\) by dividing both sides by \(-4\): \( y = -3 \).

Step 5: Substitute y into One of the Original Equations

Substitute \(y = -3\) back into the first original equation: \( 4x - 7(-3) = 21 \).

Step 6: Simplify and Solve for x

Simplify the equation: \(4x + 21 = 21\). Subtract 21 from both sides to get \(4x = 0\). Divide both sides by 4 to find \(x = 0\).

Step 7: Verify the Solution

Substitute \(x = 0\) and \(y = -3\) into the second original equation to check the solution: \(-4(0) + 3(-3) = -9\), which simplifies correctly to \(-9 = -9\), verifying the solution is correct.

Key Concepts

Substitution Method

The substitution method is one of the techniques used for solving systems of linear equations. It involves expressing one of the variables in terms of the other from one equation and then substituting this expression into the other equation. This step-by-step process allows you to reduce the system to a single equation with one variable, which is then straightforward to solve. Here’s how it works:

• Pick an Equation: Start by selecting the simpler of the equations and solve for one variable in terms of the other. This step makes subsequent substitutions easier.
• Substitute: Insert or replace the expression obtained for the variable in the other equation. This results in a new equation, which contains only one variable.
• Solve and Back-solve: Solve the single-variable equation. Then insert the obtained value into one of the original equations to find the other variable.

Using substitution is particularly straightforward when one of the equations is easily solvable for one of the variables, such as when a variable has a coefficient of 1 or -1. It is less ideal when this isn't the case, as this can lead to more complex calculations. Nonetheless, understanding this method helps build a solid foundation in algebra.

Elimination Method

The elimination method, also known as the addition method, helps eliminate one of the variables by adding or subtracting equations. Unlike substitution, this method focuses on eliminating variables first and solving for others next. This is beneficial when coefficients are conveniently set up for elimination, as demonstrated in the original solution. Here's how you apply the elimination method:

• Equalize Coefficients: Find and arrange the two equations such that the coefficients of one variable are either equal or negatives of each other.
• Add/Subtract Equations: Add or subtract the equations to get rid of that variable, resulting in an equation with one variable.
• Back-Solve: Once you’ve solved for one variable, substitute it back into one of the original equations to solve for the other.

The original exercise made the elimination method a prudent choice. By having one variable with coefficients of opposite signs, it was straightforward to solve by adding the equations together. Mastery of this technique provides flexibility and efficiency, especially with systems that align naturally for elimination.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation forms a straight line when graphed and is typically in the form of `ax + by = c`. These equations can be solved using various methods such as substitution and elimination, as seen previously. Here are some key aspects of solving linear equations:

• Identify Type: Recognize whether you have a system of linear equations (more than one equation) or a single linear equation to solve.
• Simplification: Simplify each equation if needed by combining like terms or using basic arithmetic operations.
• Choose a Method: Decide whether substitution or elimination is the best strategy given the coefficients and constants. Sometimes, balancing the simplicity of arithmetic operations can play a big role in picking the right method.
• Check and Verify: After finding solutions, it's important to substitute values back into the original equations to verify accuracy.

Linear equations form the basis for more complex algebraic structures, so honing skills in these ''solving'' strategies is essential for tackling complex mathematical problems in future coursework.