Problem 13 Write a system of equations for ... [FREE SOLUTION] (2024)

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Chapter 4: Problem 13

Write a system of equations for each problem, and then solve the system.The two domestic top-grossing movies of 2009 were Avatar and Transformers 2:Revenge of the Fallen. The movie Transformers 2 grossed \(\$ 26.9\) million lessthan Avatar, and together the two films took in \(\$ 831.1\) million. How muchdid each of these movies earn?

Short Answer

Expert verified

Avatar: \$429 million, Transformers 2: \$402.1 million

Step by step solution

01

- Define Variables

Let \( x \) be the amount of money (in millions) that Avatar grossed and let \( y \) be the amount of money (in millions) that Transformers 2 grossed.

02

- Establish Equations

From the given information, two equations can be formed:1. \( y = x - 26.9 \) (Transformers 2 grossed \$26.9 million less than Avatar)2. \( x + y = 831.1 \) (Together, the two films took in \$831.1 million)

03

- Substitute Equation

Substitute \( y = x - 26.9 \) from the first equation into the second equation:\( x + (x - 26.9) = 831.1 \)

04

- Simplify and Solve for \( x \)

Combine like terms to solve for \( x \):\[ 2x - 26.9 = 831.1 \]Add 26.9 to both sides:\[ 2x = 858 \]Divide both sides by 2:\[ x = 429 \]So, Avatar grossed \( \$429 \) million.

05

- Solve for \( y \)

Use the equation \( y = x - 26.9 \) and substitute \( x = 429 \):\[ y = 429 - 26.9 \]\[ y = 402.1 \]So, Transformers 2 grossed \( \$402.1 \) million.

Key Concepts

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

algebraic equations

An algebraic equation is a mathematical statement that shows the equality of two expressions. In this problem, we used equations to represent the relationships between the earnings of two movies, Avatar and Transformers 2.
To set up algebraic equations:

  • First, define your variables to represent unknown quantities. Here, let x be the earnings of Avatar and y be the earnings of Transformers 2.
  • Next, form equations based on the relationships described in the problem. For instance, 'Transformers 2 grossed \(26.9 million less than Avatar' translates to y = x - 26.9.
  • Another equation, 'Together they made \)831.1 million' translates to x + y = 831.1.

These algebraic equations will help us systematically determine the earnings of each movie.

substitution method

The substitution method is a technique for solving a system of linear equations. It involves solving one equation for one variable and then substituting this expression into another equation.
Here's how to use the substitution method:

  • Take the equation y = x - 26.9. Since this equation is already solved for y, we can substitute x - 26.9 in place of y in the other equation.
  • This gives us x + (x - 26.9) = 831.1. By substituting, we now only have a single variable, x, to solve for.
  • Simplify the new equation by combining like terms.

Substitution helps transform a system of equations into a more manageable single-variable equation.

solving for variables

Solving for variables means isolating the unknowns in an equation to determine their values. In a system of linear equations, this often involves algebraic manipulation.
Let's solve for x:

  • Start with the equation 2x - 26.9 = 831.1.
  • Add 26.9 to both sides to isolate terms that include x on one side, giving 2x = 858.
  • Divide both sides by 2 to find x: x = 429.

Now we know Avatar grossed \(429 million.
Next, solve for y using our earlier relation:

  • y = x - 26.9
  • Substitute x = 429 into this equation: y = 429 - 26.9 = 402.1.

So, Transformers 2 grossed \)402.1 million.

linear equations

Linear equations are equations that form straight lines when graphed. They typically take the form ax + by = c, where a, b, and c are constants.
In this problem, both our equations are linear:

  • The first equation, y = x - 26.9, represents a line where the slope is 1 and the y-intercept is -26.9.
  • The second equation, x + y = 831.1, can be rearranged to standard form as x + y - 831.1 = 0.

Solving such linear equations helps in finding precise values for unknown variables. These mathematical tools are essential in various fields like economics, physics, engineering, and more.
With a clear understanding of linear equations, we can address many practical problems with methodical, step-by-step solutions.

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Problem 13 Write a system of equations for ... [FREE SOLUTION] (3)

Most popular questions from this chapter

System of linear equations can be used to model the cost and the revenue of abusiness. You decide to sell each bike for \(\$ 600 .\) Write an equation using \(y_{2}\)(in dollars) to express your revenue when you sell \(x\) bikes.If \(x\) units of a product cost \(C\) dollars to manufacture and earn revenue of\(R\) dollars, the value of \(x\) at which the expressions for \(C\) and \(R\) areequal is called the break-even quantity-the number of units that produce 0profit. (a) find the break-even quantity, and (b) decide whether the product should beproduced on the basis of whether it will earn a profit.(Profit\(=\)Revenue\(-\)cost)\(C=105 x+6000 ; R=255 x\) No more than 400 units can be sold.Simplify. $$ (-6 x+8 y)+(6 x+2 y) $$
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Problem 13 Write a system of equations for ... [FREE SOLUTION] (2024)

FAQs

How do you write a system of equations with no solution? ›

A system of two linear equations has no solution if the lines are parallel. Parallel lines on a coordinate plane have the same slope and different y-intercepts (see figure 3 for an example of this). If the lines look parallel, confirm it by checking that they have the same slope.

What is the solution answer to a system of equations? ›

The solution set to a system of equations will be the coordinates of the ordered pair(s) that satisfy all equations in the system. In other words, those values of x and y will make the equations true. Accordingly, when a system of equations is graphed, the solution will be all points of intersection of the graphs.

What is the process of elimination in math? ›

HOW TO SOLVE A SYSTEM OF EQUATIONS BY ELIMINATION.
  1. Write both equations in standard form. ...
  2. Make the coefficients of one variable opposites. ...
  3. Add the equations resulting from Step 2 to eliminate one variable.
  4. Solve for the remaining variable.
  5. Substitute the solution from Step 4 into one of the original equations.
Mar 3, 2024

How to solve an equation? ›

To solve any equation we need to perform arithmetic operations, to separate the variable, such as: Adding the same number on both sides. Subtracting same number on both sides. Multiplying with the same number on both sides.

What is the formula for no solution? ›

Condition for No Solution:

Considering the pair of linear equations by two variables u and v. Therefore a1, b1, c1, a2, b2, c2 are real numbers. If (a1/a2) = (b1/b2) ≠ (c1/c2), then this will result in no solution.

Is 0 0 no solution? ›

If you get an equation that is always true, such as 0 = 0, then there are infinite solutions.

What is an example of an elimination equation? ›

For example, let us solve two equations 2x - y = 4 → (1) and 4x - 2y = 7 → (2) by the elimination method. In order to make the x coefficients equal in both the equations, we multiply equation (1) by 2 and equation (2) by 1. By doing so we get, 4x - 2y = 8 → (3) and 4x - 2y = 7 → (4).

How to solve a system of linear equations? ›

SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION.
  1. Solve one of the equations for either variable.
  2. Substitute the expression from Step 1 into the other equation.
  3. Solve the resulting equation.
  4. Substitute the solution in Step 3 into either of the original equations to find the other variable.
Nov 24, 2022

What is the 4 basic rule in solving equations? ›

In order to solve an algebra equation, try to isolate the variable on one side of the equals sign by adding, subtracting, multiplying, and dividing both sides of the equation by values which simplify the equation.

What is the easiest way to solve equations? ›

The following steps provide a good method to use when solving linear equations.
  1. Simplify each side of the equation by removing parentheses and combining like terms.
  2. Use addition or subtraction to isolate the variable term on one side of the equation.
  3. Use multiplication or division to solve for the variable.
Sep 7, 2021

What are the 3 methods for solving equations? ›

So, in order to solve that problem, you need to be able to find the value of all the variables in each equation. There are three different ways that you could do this: the substitution method, elimination method, and using an augmented matrix.

How do you show an equation has no solutions? ›

Some equations have no solutions. In these equations, there is no value for the variable that makes the equation true. You can tell that an equation has no solutions if you try to solve the equation and get a false statement.

Which describes a system of equations with no solution? ›

If a system has no solution, it is said to be inconsistent .

How to tell if a system of equations has no solution or infinitely many? ›

If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions..

What is a system of lines with no solution? ›

No Solution: When the lines that make up a system are parallel, there are no solutions because the two lines share no points in common.

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