Problem 22 - Algebra 1

Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &5 e+4 f=9\\\ &4 e+5 f=9 \end{aligned}$$

Step-by-Step Solution

Answer

The solution of the system of equations is \(e = 1\) and \(f = 1\).

Step 1: Multiply both equations by suitable numbers to obtain the same coefficient for f/e

Multiplying the first equation by 4 and the second equation by 5 gives us:\[\begin{aligned} &20 e+16 f=36\ &20 e+25 f=45 \end{aligned}\]

Step 2: Subtract the second equation from the first

Subtracting these equations:\[20e+16f - (20e+25f) = 36-45\]gives us: \[-9f = -9\]

Step 3: Solve for variable f

To solve for f, we divide both sides by -9:\[f = 1\]

Step 4: Substitute f in one of the initial equations and solve for e

Substitute \(f = 1\) in the first equation \(5e+4f=9\) to solve for e:\[5e + 4(1) = 9 \Rightarrow 5e = 5 \Rightarrow e = 1\]

Key Concepts

Linear Combinations

A linear combination involves creating new equations by multiplying existing equations by constants and adding or subtracting them. This technique is particularly useful for solving systems of linear equations since it allows us to eliminate one variable at a time to make solving the system easier. In the context of our exercise, we form linear combinations of the given equations to eliminate one variable. We first identify a common coefficient for either variable \(e\) or \(f\) by multiplying each equation with a suitable number, as shown in the solution steps. This process transforms the equations into a format that allows one to easily eliminate a variable.
Linear combinations can bring clarity to complex problems and simplify solving even larger systems of equations. This flexibility allows us to manipulate equations to achieve the desired outcome, which is isolating a single variable.

System of Equations

A system of equations is simply a set of two or more equations with multiple variables. These systems are a fundamental part of algebra that allow us to find values that satisfy all equations simultaneously. In our exercise, we have a system consisting of two linear equations. Each equation represents a line, and our task is to find the point(s) where these lines intersect, which would be the solution to the system.

• Two equations can represent intersecting, parallel, or coincident lines.
• Their solutions might be one point, no solution, or infinitely many solutions, respectively.

Using the system, we aim to find the values of \(e\) and \(f\) that satisfy both equations. Successful solving ensures that these values make both equations true when substituted back.

Solving Equations

Solving equations often involves isolating the variable of interest, either by linear combinations, substitution, or other algebraic methods like elimination. In the given system, once we used a linear combination to eliminate \(e\), we were left with a single variable equation in \(f\). This straightforward equation \(-9f = -9\) can be easily solved by simplifying and dividing both sides by the coefficient of \(f\).
Finally, we substitute the found value back into one of the original equations to determine the other variable, \(e\). This step ensures that both values satisfy the entire system. Solving equations systematically and checking your solutions by substituting them back into the original equations is critical in verifying the correctness of the results.