Question

Orbital Launches In 2017 there was a total of 469 commercial and noncommercial orbital launches worldwide. In addition, the number of noncommercial orbital launches was 31 more than half the number of commercial orbital launches. Determine the number of commercial and noncommercial orbital launches in \(2017 .\)

Short answer

292 commercial launches and 177 noncommercial launches

Step-by-step solution

Define Variables

Let the number of commercial orbital launches be denoted as \(x\) and the number of noncommercial orbital launches be denoted as \(y\).

Set Up Equations

From the problem, we have two equations: 1) The total number of launches is 469:\[ x + y = 469 \]2) The number of noncommercial launches is 31 more than half the number of commercial launches:\[ y = \frac{1}{2} x + 31 \]

Substitute \(y\) from the Second Equation into the First Equation

Substitute \( y = \frac{1}{2} x + 31 \) into the first equation:\[ x + (\frac{1}{2} x + 31) = 469 \]

Simplify and Solve for \(x\)

Combine like terms and solve for \(x\):\[ x + \frac{1}{2} x + 31 = 469 \]\[ \frac{3}{2} x + 31 = 469 \]Subtract 31 from both sides:\[ \frac{3}{2} x = 438 \]Multiply both sides by \( \frac{2}{3} \) to solve for \(x\):\[ x = 292 \]

Solve for \(y\)

Use the value of \(x\) to find \(y\):\[ y = \frac{1}{2} (292) + 31 \]\[ y = 146 + 31 \]\[ y = 177 \]

Verify the Solution

Check that the total number of launches adds up to 469 and that the number of noncommercial launches is 31 more than half the number of commercial launches:\[ x + y = 292 + 177 = 469 \]And\[ y = \frac{1}{2} x + 31 \implies 177 = \frac{1}{2} (292) + 31 \]

Key concepts

linear equations

Linear equations are mathematical statements that show the relationship between two variables in a straight line when graphed. They usually take the form\( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. Linear equations are straightforward and can usually be solved in a few steps. For instance, if you know that the total number of commercial and noncommercial orbital launches is 469, you can write a linear equation like \( x + y = 469 \), where \( x \) is commercial launches and \( y \) is noncommercial launches.
Linear equations are powerful tools for solving real-world problems, such as determining how many orbital launches took place in a given year.

substitution method

The substitution method is a technique used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method simplifies the system into a single equation with just one variable.
For instance, if you have the equations \( x + y = 469 \) and \( y = \frac{1}{2} x + 31 \), you can solve the second equation for y and then substitute that expression into the first equation:

• Start with \( y = \frac{1}{2} x + 31 \).
• Substitute this into the first equation: \( x + (\frac{1}{2} x + 31) = 469 \).
• Simplify and solve for \( x \).

The substitution method is particularly useful when one equation is already solved for one variable.

variable definition

In solving systems of linear equations, defining your variables is crucial. Variables are symbols that represent unknown values in equations. Clear and consistent variable definitions make it easier to set up and solve equations.
For instance, in this problem, we defined:

• \( x \) as the number of commercial orbital launches.
• \( y \) as the number of noncommercial orbital launches.

Setting up these variable definitions helps us create understandable equations. Without proper variable definitions, it can be hard to follow the steps needed to solve the system of equations.

problem-solving steps

When solving a system of linear equations, follow these clear steps to ensure success:

1. Define your variables. For example, let \( x \) be the number of commercial orbital launches and \( y \) be the number of noncommercial orbital launches.
2. Set up your equations based on the given problem. Use information from the problem to form equations like \( x + y = 469 \) and \( y = \frac{1}{2} x + 31 \).
3. Use the substitution method or another solving technique. For example, substitute \( y = \frac{1}{2} x + 31 \) into \( x + y \).
4. Solve the resulting simplified equation. For example, solving \( x + \frac{1}{2} x + 31 = 469 \) leads to \( x = 292 \).
5. Substitute back to find the other variable. Using \( x = 292 \), calculate y as \( \frac{1}{2} (292) + 31 = 177 \).
6. Verify your results. Check if \( x + y = 469 \) and \( y = \frac{1}{2} x + 31 \) hold true.

By breaking down the problem into these manageable steps, you can carefully solve the system of linear equations.