Problem 40 - Precalculus with Limits

Question

Solving a system In Exercises \(35-40,\) use any method to solve the system. $$ \left\\{\begin{aligned} 4 x-3 y &=6 \\\\-5 x+7 y &=-1 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solutions to the system of equations are \(x = 3\) and \(y = 2\).

Step 1: Multiply each equation by a suitable factor

In order to use the elimination method, both equations should be multiplied by suitable factors to make the coefficient of y the same in each equation(or x). So multiplying the first equation by 7 and the second by 3 we get: \(28x - 21y = 42\) and \(-15x + 21y = -3\)

Step 2: Add the two equations

When both equations are added, the y-terms cancel out, resulting in: \(13x = 39\)

Step 3: Solve for x

By dividing both sides of the equation by 13, the value of x can be found: \(x = 3\)

Step 4: Substitute the value of x in the first equation

By substituting \(x = 3\) into the first equation \(4x - 3y = 6\), we get: \(12 - 3y = 6\)

Step 5: Solve for y

By rearranging and dividing by -3, the value for y can be found: \(y = 12 - 6 / -3 = 2\)

Key Concepts

Elimination Method

The elimination method is a traditional and effective strategy to solve systems of linear equations. This method involves removing one variable to make it easier to solve for the other. Essentially, by adding or subtracting equations, one of the variables is eliminated, simplifying the system significantly.

To use the elimination method effectively:

• Multiply one or both of the equations by a number that will allow you to cancel out one of the variables when the equations are combined.
• Make sure the coefficient of one of the variables is the same in both equations, but with opposite signs.
• Add or subtract the equations from each other so that one variable is eliminated.
• Solve the resulting single-variable equation.

This approach works well for larger systems and can even be extended to more complex equations.

Substitution Method

The substitution method is another way to solve systems of equations, and it works by solving one of the equations for one variable in terms of the others. Once isolated, this variable is then substituted back into the other equation.

The steps to solve using substitution include:

• Choose one of the equations and solve it for one of its variables.
• Take that expression and substitute it into the other equation.
• This substitution will give you an equation with only one variable remaining.
• Solve this equation to find the value of the first variable.
• Finally, use this value to solve for the other variable.

This method is particularly useful when one equation is easily solvable for a single variable. It's a great fit when the methods of direct comparison or graphing don't offer simplicity.

Solving Systems of Equations

Solving systems of equations refers to finding the values for variables that satisfy all equations in the system simultaneously. This process is crucial in both everyday problem solving and advanced mathematics.

There are several methods to solve such systems:

• Elimination Method: Combines equations to eliminate a variable and solve the system more easily.
• Substitution Method: Replaces one variable with an equivalent expression to solve the system step-by-step.
• Graphing Method: Involves plotting each equation on a graph to find intersection points; it can be visually intuitive but less precise for complex equations.

Choosing the right method often depends on the specific problem and the form of the equations. Each technique has its strengths and weaknesses, and understanding how to apply them is key to mastering systems of equations.