Problem 34 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{aligned} &7 x-4 y=-1\\\ &3 x-5 y=16 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = -3\) and \(y = -5\).

Step 1: Align the Equations

The given system of equations is:\[ \begin{aligned} 7x - 4y &= -1 \ 3x - 5y &= 16 \end{aligned} \] We need to eliminate one of the variables by making the coefficients equal in magnitude and opposite in sign.

Step 2: Make Coefficients of y Equal

To eliminate \(y\), we need to equalize the coefficients of \(y\) in both equations. The coefficients are \(-4\) and \(-5\). We can multiply the first equation by \(5\) and the second equation by \(4\) to achieve this:\[ \begin{aligned} 5(7x - 4y) &= 5(-1) \ 4(3x - 5y) &= 4(16) \end{aligned} \]This leads to:\[ \begin{aligned} 35x - 20y &= -5 \ 12x - 20y &= 64 \end{aligned} \]

Step 3: Subtract the Equations

Subtract the second equation from the first to eliminate \(y\):\[ (35x - 20y) - (12x - 20y) = -5 - 64 \]Simplifying,\[ 35x - 12x = -5 - 64 \ 23x = -69 \]

Step 4: Solve for x

Solve the equation \(23x = -69\) for \(x\):\[ x = \frac{-69}{23} \ x = -3 \]

Step 5: Substitute x into Original Equation

Substitute \(x = -3\) back into one of the original equations (use the first equation for simplicity):\[ 7(-3) - 4y = -1 \ -21 - 4y = -1 \]

Step 6: Solve for y

Simplify and solve the equation for \(y\):\[ -21 - 4y = -1 \ -4y = 20 \ y = -5 \]

Step 7: Write the Solution

The solution to the system of equations is:\[ x = -3, \quad y = -5 \]

Key Concepts

Linear Equations

A linear equation is a type of equation where each term is either a constant or the product of a constant and a single variable. Linear equations can look quite simple, such as \(x + 2 = 5\), or more complex, such as \(7x - 4y = -1\).

Despite their complexity, they all graph as straight lines when plotted. This is what makes them 'linear'.

There are no squared terms or variables multiplied together.

• They usually take the form \(ax + by = c\).
• They can have one or more variables, like \(x\) and \(y\) in our example.
• Solutions to linear equations provide the values of these variables that make the equation true.

Understanding the characteristics of linear equations is essential when solving them.

System of Equations

A system of equations is a set of two or more equations with the same variables. In our exercise, the system includes two equations: - \(7x - 4y = -1\)- \(3x - 5y = 16\)

These equations are solved simultaneously. This means we are trying to find values for \(x\) and \(y\) that work in both equations at the same time.

The goal is to find a solution that satisfies all equations in the system. Here are some key points:

• Systems can have one solution (a single point), infinitely many solutions (all points on a line), or no solution (parallel lines).
• Their solutions can often be visualized as the intersection of the lines represented by each equation.
• There are several methods to solve them, including graphing, substitution, and elimination.

Understanding systems of equations is vital to learning how to solve them correctly.

Solving Equations

Solving equations involves finding the value of variables that make an equation true. When solving, especially in the context of a system, following a series of logical steps ensures that you accurately find the solutions.

Steps to solving equations typically include:

• Simplifying each equation, clearing fractions or parentheses if necessary.
• Isolating one variable, to find its relationship with others or with constants.
• Substituting known values from one step into another equation.

For systems of linear equations, the objective is to find values for every variable such that all given equations are true simultaneously.

Regular practice of these steps builds confidence and accuracy in finding solutions to various types of equations.

Variables Elimination

The elimination method is used to remove one variable from a system of equations, allowing us to solve for the other variable more easily. In our exercise, we used elimination to find \(x\) and \(y\):

The key idea in elimination is to make the coefficients of one variable the same and then subtract or add equations to cancel that variable.

Steps to eliminate variables include:

• Equalize the coefficients of the variable you want to eliminate across all equations.
• Subtract or add the equations to cancel out the desired variable, resulting in a single-variable equation.
• Solve the resulting equation for one variable, and substitute back to find the other variable.

This method simplifies the process of solving complex systems, making it a very powerful tool for students.