Chapter 13: Problem 11
Jacob is now 12 years younger than Michael. If 9 years from now Michael will be twice as old as Jacob, how old will Jacob be in 4 years?
Short Answer
Expert verified
Jacob will be 7 years old in 4 years.
Step by step solution
01
- Define the Variables
Let J be Jacob's current age and M be Michael's current age.
02
- Set Up the Equations
From the problem, we know that Jacob is 12 years younger than Michael. This relationship can be written as the equation: J = M - 12
03
- Set Up the Future Age Condition
Nine years from now, Michael will be twice as old as Jacob. This gives us the equation: M + 9 = 2(J + 9)
04
- Substitute Equation Into Future Age Condition
Substitute the expression for J from the first equation (J = M - 12) into the second equation: M + 9 = 2((M - 12) + 9)
05
- Simplify and Solve
Simplify and solve for M: M + 9 = 2(M - 3) M + 9 = 2M - 6 15 = M
06
- Find Jacob's Current Age
Substitute M = 15 back into the equation J = M - 12: J = 15 - 12 J = 3
07
- Determine Jacob's Age in 4 Years
Jacob is currently 3 years old. In 4 years, Jacob will be: 3 + 4 = 7
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
age-related algebra problems
Age-related algebra problems often involve comparing the ages of two or more people using mathematical equations.
In these problems, you usually have to identify the relationship between their ages and then solve for one or more unknown values.
This type of problem is common because it helps improve your skills in setting up and solving linear equations.
Age-related problems typically start with some known information, like one person being a certain number of years older or younger than another.
Using this information, you can form equations to represent the given relationships.
For instance, if Jacob is 12 years younger than Michael, we write it as: \(J = M - 12\).
From here, you solve the problem by creating more equations based on future conditions, such as ages at different times.
In these problems, you usually have to identify the relationship between their ages and then solve for one or more unknown values.
This type of problem is common because it helps improve your skills in setting up and solving linear equations.
Age-related problems typically start with some known information, like one person being a certain number of years older or younger than another.
Using this information, you can form equations to represent the given relationships.
For instance, if Jacob is 12 years younger than Michael, we write it as: \(J = M - 12\).
From here, you solve the problem by creating more equations based on future conditions, such as ages at different times.
solving linear equations
Linear equations are fundamental in algebra and represent relationships that form straight lines when graphed.
In the context of age-related problems, you often solve linear equations to find unknown ages.
Usually, you will have one or more equations that you need to solve simultaneously.
Let’s use our current example where we have \( J = M - 12 \) and \( M + 9 = 2(J + 9) \).
You substitute and simplify these equations to find the values of J and M.
The process involves moving all variable terms to one side of the equation and constants to the other side, then solving for the variable.
In solving linear equations, you might encounter different methods such as substitution or elimination, which we will cover in further sections.
In the context of age-related problems, you often solve linear equations to find unknown ages.
Usually, you will have one or more equations that you need to solve simultaneously.
Let’s use our current example where we have \( J = M - 12 \) and \( M + 9 = 2(J + 9) \).
You substitute and simplify these equations to find the values of J and M.
The process involves moving all variable terms to one side of the equation and constants to the other side, then solving for the variable.
In solving linear equations, you might encounter different methods such as substitution or elimination, which we will cover in further sections.
substitution method
The substitution method is a common technique in algebra for solving systems of equations.
The basic idea is to solve one equation for one variable and then substitute that expression into another equation.
In our age-related problem, you have two equations: \( J = M - 12 \) and \( M + 9 = 2(J + 9) \).
First, solve the first equation for J: \( J = M - 12 \).
Next, replace J in the second equation with \( M - 12 \): \( M + 9 = 2((M - 12) + 9) \).
This step helps reduce the number of variables and simplifies the problem into a single linear equation which can be easily solved.
After substituting and simplifying, you can solve for the remaining variable.
The basic idea is to solve one equation for one variable and then substitute that expression into another equation.
In our age-related problem, you have two equations: \( J = M - 12 \) and \( M + 9 = 2(J + 9) \).
First, solve the first equation for J: \( J = M - 12 \).
Next, replace J in the second equation with \( M - 12 \): \( M + 9 = 2((M - 12) + 9) \).
This step helps reduce the number of variables and simplifies the problem into a single linear equation which can be easily solved.
After substituting and simplifying, you can solve for the remaining variable.
variable definition
Defining variables is an essential first step in solving any algebra problem.
It involves assigning variables to unknown quantities you want to find.
For example, in age-related problems, you might define J as Jacob's current age and M as Michael's current age.
This simple definition helps to translate the word problem into algebraic equations.
Once you define the variables, you can set up equations based on the relationships described in the problem.
In our example, we wrote \( J = M - 12 \) to denote that Jacob is 12 years younger than Michael.
Correctly defining your variables and using them consistently throughout the problem makes the solution process much smoother and more understandable.
It involves assigning variables to unknown quantities you want to find.
For example, in age-related problems, you might define J as Jacob's current age and M as Michael's current age.
This simple definition helps to translate the word problem into algebraic equations.
Once you define the variables, you can set up equations based on the relationships described in the problem.
In our example, we wrote \( J = M - 12 \) to denote that Jacob is 12 years younger than Michael.
Correctly defining your variables and using them consistently throughout the problem makes the solution process much smoother and more understandable.
mathematical reasoning
Mathematical reasoning is crucial for understanding and solving algebra problems effectively.
It involves logical thinking and the ability to translate a word problem into mathematical equations.
In the given problem, reasoning helps identify relationships between ages and future conditions to form equations.
For instance, knowing that Michael's age in the future will be twice Jacob’s age leads to the equation \( M + 9 = 2(J + 9) \).
Once you establish your equations, reasoning continues to play a role as you simplify and solve these equations.
Understanding each step and why it makes sense helps in applying the same techniques to solve different problems.
Also, checking your final solution in the context of the original problem ensures that the result is logical and correct.
It involves logical thinking and the ability to translate a word problem into mathematical equations.
In the given problem, reasoning helps identify relationships between ages and future conditions to form equations.
For instance, knowing that Michael's age in the future will be twice Jacob’s age leads to the equation \( M + 9 = 2(J + 9) \).
Once you establish your equations, reasoning continues to play a role as you simplify and solve these equations.
Understanding each step and why it makes sense helps in applying the same techniques to solve different problems.
Also, checking your final solution in the context of the original problem ensures that the result is logical and correct.
