# An exam worth 378 points contains 66 questions. Some questions are worth 7 points, and the others are worth 4 points. How many 7 point and 4 point questions are on the test?

##### 2 Answers

There are 38 7 point questions and 28 4 point questions.

#### Explanation:

This is a problem that requires two equations since there are two variables.

The first equation is easy it is the total number of questions.

x + y = 66 where x = 7 point questions and y = 4 point questions

The second equation is the total number of points.

7(x) + 4(y) = 378 7x is the points from the 7 point questions.

4y is the points from the 4 point questions.

378 is the total number of points.

To solve the two equations substitute the values from the first equation into the second equation.

x + y = 66 subtract x from both sides

x - x + y = 66 - x which gives

```
y = ( 66 - x ) now substitute the value into the second equation.
in place of y.
```

7x +4 ( 66-x) = 378 Now there is an equation with only one variable.

Now multiply 4 across the parenthesis using the distributive property. This gives

7x + 264 - 4x = 378 now combine the x values using the associate property this results in

7x - 4x + 264 = 378 Subtract the 4x from the 7x to get

3x + 264 = 378 now subtract 264 from both sides

3x + 264 - 264 = 378 - 264 which results in

3x = 114 divide both sides by 3

x = 38 Now to solve for y put x back into the first equation

38 + y = 66 Subtract 38 from both sides.

38 - 38 + y = 66 - 38 which gives

```
y = 28
```

So the 7 point questions = 38 and

the 4 point questions = 28

To solve a problem with two variables requires two equations.

Then solve one equation for one of the variables and substitute this value into the second equation.

#### Explanation:

This is a system of linear equations question

Let

Let

Total Number of question equation

Total Point value equation

We have several methods that could be used

1)Graphing

2)Elimination

3)Substitution

4)Matrices (I did not include this method yet)

**Graphing**

Use a graphing tool and input the 2 equations. Look for where the 2 equations intersect. The intersection point (the ordered pair) is the solution to the problem.

**Elimination Method:**

We need to eliminate one of the variables. Let's eliminate the

Simplify

Now add this equation with the other equation to eliminate the

Now isolate the

Now substitute this value for

Subtract 38 from both sides

**Substitution Method:**

Take either equation and solve for either variable. We will take the first equation because it is the easiest to work with. We will solve for

Subtract x from both sides

y=66-x#

Now substitute this into the second equation

Distribute

Combine like terms

Subtract 264 from both sides

Simplify

Isolate

Now substitute this value for

Subtract 38 from both sides

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