Question

The sum of two numbers is \(16 .\) If the larger number is 4 less than 3 times the smaller number, find the two numbers.

Short answer

The two numbers are 5 and 11.

Step-by-step solution

Define Variables

Let the smaller number be denoted as \( x \) and the larger number be denoted as \( y \).

Set Up Equations

Based on the problem statement, create the following equations: 1. \( x + y = 16 \) (Sum of the two numbers is 16) 2. \( y = 3x - 4 \) (Larger number is 4 less than 3 times the smaller number)

Substitute and Solve for Smaller Number

Substitute \( y \) from the second equation into the first equation: \( x + (3x - 4) = 16 \)Simplify and solve for \( x \): \( 4x - 4 = 16 \)\( 4x = 20 \)\( x = 5 \)

Solve for Larger Number

Use the value of \( x \) to find \( y \): \( y = 3(5) - 4 \) \( y = 15 - 4 \) \( y = 11 \)

Conclusion

The smaller number is \( 5 \) and the larger number is \( 11 \).

Key concepts

Simultaneous Equations

Simultaneous equations are multiple equations that involve two or more variables. The solution to these equations is the set of values for the variables that satisfy all equations at the same time. In the given exercise, we have two equations:

1. \( x + y = 16 \)
2. \( y = 3x - 4 \)

The goal is to find the values of \( x \) and \( y \) that make both of these equations true simultaneously.

Variable Substitution

Variable substitution is a method used to simplify simultaneous equations. We replace one variable in one equation with an expression involving the other variable. This turns a system of equations into a single equation we can solve more easily.

In our problem, the second equation tells us \( y = 3x - 4 \). We substitute this expression for \( y \) in the first equation:

\( x + (3x - 4) = 16 \).

Now we have a single equation with one variable.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they involve only the first power of the variable. After substitution, our equation is:

\( x + 3x - 4 = 16 \)

Simplify this to get:

\( 4x - 4 = 16 \)

Next, isolate \( x \) by adding 4 to both sides:

\( 4x = 20 \)

Finally, divide by 4 to solve for \( x \):

\( x = 5 \).

Now we know the value of the smaller number.

Problem-Solving

Solving mathematical problems like this one involves several important steps:

• Define the variables clearly.
• Set up equations based on the problem statement.
• Use methods like variable substitution to simplify the equations.
• Solve the simplified equations methodically.
• Substitute back to find any remaining unknowns.
• Check your solution to ensure it meets all conditions of the problem.

In our case, after finding \( x = 5 \), we substitute it back into \( y = 3x - 4 \) to get:

\( y = 3(5) - 4 = 15 - 4 = 11 \).

Thus, our final solution is that the smaller number is 5 and the larger number is 11.