Your math teacher tells you that the next test is worth 100 points and contains 38 problems. multiple choice questions are worth 2 points each and word problems are 5 points. how many of each type are there

30 multiple choice questions and 8 word problems

Step-by-step explanation:

Let m = number of multiple choice questions.

Let w = number of word problems.

There are 38 problems altogether, so we have the first equation concerning the number of problems.

m + w = 38

A multiple choice question is worth 2 points, so m multiple choice questions are worth 2m points. A word problem is worth 5 points, so w word problems are worth 5w points. Altogether, both types of problems are worth 2m + 5w points. We are told all problems combined are worth 100 points. We can now write our second equation concerning the number of points.

2m + 5w = 100

We have a system of two equations in two unknowns.

m + w = 38

2m + 5w = 100

We can use the substitution method to solve the system of equations.

Solve the first equation for m.

m + w = 38

Subtract w from both sides.

m = 38 - w

Now where we see m in the second original equation, we substitute it with 38 - w.

The second original equation is

2m + 5w = 100

Substitute 38 - w for m in the equation.

2(38 - w) + 5w = 100

We have a single equation in a single variable, so we can solve it for the variable.

Distribute the 2.

76 - 2w + 5w = 100

Combine like terms on the left side.

76 + 3w = 100

Subtract 76 from both sides.

3w = 24

Divide both sides by 3.

w = 8

There are 8 word problems.

Now we use the first original equation and substitute 8 for w and solve for m.

m + w = 38

m + 8 = 38

Subtract 8 from both sides.

m = 30

There are 30 multiple choice questions.

30 multiple choice questions and 8 word problems

Check:

30 questions + 30 problems = 38 problems in total

30 questions @ 2 pts each = 2 * 30 pts = 60 pts

8 problems @ 5 pts each = 5 * 8 pts = 40 pts

60 pts + 40 pts = 100 pts

The total is 38 problems and 100 points, so our answer is correct.

1^3 + 2^3 + 3^3 +4^3 = (1+2+3+4)^2

step-by-step explanation:

i am only able to answer the first part. i am not sure about the second part. sorry