Jim has a total of 77 red and blue marbles. The number of blue marbles is five more than twice the number of red marbles.
Write a pair of linear equations to represent the information. Be sure to state what the variables represent. (2 points)
Explain the substitution method of solving this pair of equations. (2 points)
Solve the equations to find the number of red marbles. Show your work. (3 points)
Let's let "r" represent the number of red marbles and "b" represent the number of blue marbles. Then we can create two equations based on the given information:
r + b = 77 (because Jim has a total of 77 marbles)
b = 2r + 5 (because the number of blue marbles is five more than twice the number of red marbles)
To solve this system of equations using substitution, we want to solve one of the equations for one of the variables, and then substitute that expression into the other equation. For example, we could solve the second equation for "b":
b = 2r + 5
Then, we can substitute this expression for "b" into the first equation:
r + (2r + 5) = 77
Now we have an equation with just one variable (r) that we can solve:
3r + 5 = 77
3r = 72
r = 24
So Jim has 24 red marbles. To find the number of blue marbles, we can substitute this value into either of the original equations:
b = 2r + 5
b = 2(24) + 5
b = 53
So Jim has 24 red marbles and 53 blue marbles.
Let's assign variables to represent the unknown quantities in the problem. Let's say 'r' represents the number of red marbles and 'b' represents the number of blue marbles.
Based on the given information, we can write the following equations:
Equation 1: r + b = 77
This equation represents the total number of marbles, which is 77.
Equation 2: b = 2r + 5
This equation represents the relationship between the number of blue marbles and twice the number of red marbles, with an additional 5 blue marbles.
To solve this pair of equations using the substitution method, we follow these steps:
1. Solve one equation for one variable in terms of the other variable. In this case, Equation 2 is already solved for 'b' in terms of 'r', so we can substitute this expression for 'b' in Equation 1.
2. Substitute the expression found in step 1 into the other equation.
3. Solve the resulting equation for one variable.
4. Substitute the value found in step 3 back into one of the original equations to find the value of the other variable.
Now, let's solve the equations to find the number of red marbles:
Step 1:
b = 2r + 5
Step 2:
Substituting '2r + 5' for 'b' in Equation 1:
r + (2r + 5) = 77
Step 3:
Combine like terms:
3r + 5 = 77
Subtract 5 from both sides:
3r = 72
Divide both sides by 3:
r = 24
Step 4:
Substitute 'r = 24' into Equation 2:
b = 2(24) + 5
b = 48 + 5
b = 53
Therefore, the number of red marbles is 24.