Problem 32 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{aligned} &5 x-2 y=7\\\ &3 x+5 y=29 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 3\) and \(y = 4\).

Step 1: Align the equations

First, write down the given system of equations clearly: \(\begin{aligned}5x - 2y &= 7 \3x + 5y &= 29\end{aligned}\)

Step 2: Make coefficients equal

Look to eliminate one variable by making the coefficients of either \(x\) or \(y\) equal in both equations. Let's eliminate \(y\), so we multiply the first equation by 5 and the second by 2.Equation 1 becomes: \(5(5x - 2y) = 5(7)\) which simplifies to \(25x - 10y = 35\)Equation 2 becomes: \(2(3x + 5y) = 2(29)\) which simplifies to \(6x + 10y = 58\).

Step 3: Add the equations

Now add these two newly formed equations to eliminate \(y\): \(\begin{aligned}25x - 10y &= 35 \6x + 10y &= 58 \------------&-----------\31x &= 93\end{aligned}\)

Step 4: Solve for x

Divide the equation \(31x = 93\) by 31 to solve for \(x\):\(x = \frac{93}{31} = 3\).

Step 5: Substitute back to find y

With \(x = 3\), substitute this value back into one of the original equations to solve for \(y\). Using \(5x - 2y = 7\):\(\begin{aligned}5(3) - 2y &= 7 \15 - 2y &= 7 \-2y &= 7 - 15 \-2y &= -8 \y &= 4\end{aligned}\)

Step 6: Solution

The solution to the system of equations is \(x = 3\) and \(y = 4\). You can check this solution by substituting \(x\) and \(y\) back into both original equations to ensure both hold true.

Key Concepts

Elimination Method

The elimination method is a popular technique in solving systems of linear equations. It involves manipulating the equations such that one variable is eliminated, making it easier to solve for the other variable.
By focusing on one variable at a time, it simplifies the problem. Here's how it works:

• Choose which variable to eliminate. Often, it's easiest to choose the one with coefficients that can be easily matched or canceled out when multiplied.
• Adjust the coefficients of the chosen variable in each equation so that they can cancel each other out when the equations are added or subtracted.
• Add or subtract the equations to eliminate one of the variables.
• Solve the resulting single-variable equation.

This method reduces the complexity of dealing with two variables simultaneously, streamlining the solution process.

Solving Equations

Solving equations, particularly when dealing with systems of equations, involves finding such values for the variables that make the equation truly balanced. Each variable represents an unknown, and the goal is to find the exact numbers that satisfy the given conditions.For linear equations:

• Identify your unknowns, typically represented by letters such as \(x\) and \(y\).
• Use algebraic techniques like addition, subtraction, substitution, or elimination to isolate and discover these unknowns.
• Always ensure that every manipulated equation remains equivalent to the original to avoid errors.

When you have two equations like in this exercise, solving them simultaneously means finding a pair of values that makes both equations true at the same time. This requires precision and a methodical approach to ensure all calculations are correct.

Linear Algebra

Linear Algebra is a branch of mathematics that focuses on systems of linear equations, vectors, matrices, and more. It's essential for understanding how to handle linear systems and is widely applied in various fields such as computer science, engineering, and economics. In relation to solving systems of linear equations:

• Linear equations are equations where each term is either a constant or the product of a constant and a single variable. They graph as straight lines.
• Systems of linear equations can be solved using methods like substitution, elimination, or matrix operations.
• In Linear Algebra, these equations can also be represented and solved using matrix operations, offering a robust technique for handling larger systems.

Ultimately, understanding Linear Algebra enhances your problem-solving skills and applies to many real-world scenarios, where predicting, optimizing, or understanding complex systems is required.