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Chapter 7: Problem 40

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

Short Answer

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

Step by step solution

01

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 \(-15x + 21y = -3\)
02

Add the two equations

When both equations are added, the y-terms cancel out, resulting in: \(13x = 39\)
03

Solve for x

By dividing both sides of the equation by 13, the value of x can be found: \(x = 3\)
04

Substitute the value of x in the first equation

By substituting \(x = 3\) into the first equation \(4x - 3y = 6\), we get: \(12 - 3y = 6\)
05

Solve for y

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Sports In Super Bowl I, on January \(15,1967\) , the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to \(10 .\) The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth \(6,1 ,\) and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. There were six times as many touchdowns as field goals. Find the numbers of touchdowns, extra-point kicks, and field goals scored. (Source: National Football League)

Investment Analysis A person plans to invest up to \(\$ 20,000\) in two different interest-bearing accounts. Each account is to contain at least \(\$ 5000 .\) Moreover, the amount in one account should be at least twice the amount in the other account. Find and graph a system of inequalities to describe the various amounts that can be deposited in each account.

Finding Minimum and Maximum Values, find the minimum and maximum values of the objective function and where they occur, subject to the constraints \(x \geq 0, y \geq 0,3 x+y \leq 15\) and \(4 x+3 y \leq 30 .\) $$ z=x+y $$

Solving a Linear Programming Problem, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=4 x+5 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+y \geq 8} \\ {3 x+5 y \geq 30}\end{array} $$

The _____ function of a linear programming problem gives the quantity to be maximized or minimized.
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