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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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