# 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
Oct 7, 2016

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

$3 \frac{x}{3} = \frac{114}{3}$ the answer is

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.

Oct 7, 2016

$38$ questions are valued at 7 points each.
$28$ questions are valued at 4 points each.

#### Explanation:

This is a system of linear equations question

Let $x$ = $7$ point questions
Let $y$ = $4$ point questions

Total Number of question equation

$x + y = 66$

Total Point value equation

$7 x + 4 y = 378$

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 $y$ variable by multiplying the first equation by $- 4$.

$\left(- 4\right) x + \left(- 4\right) y = \left(- 4\right) 66$

Simplify

$- 4 x - 4 y = - 264$

Now add this equation with the other equation to eliminate the $y$ variable.

$- 4 x - 4 y = - 264$
$7 x + 4 y = 378$

$3 x = 114$

Now isolate the $x$ variable by dividing

$\cancel{3} \frac{x}{\cancel{3}} = \frac{114}{3}$

$x = 38$

Now substitute this value for $x$ in the original version of the first equation.

$\textcolor{red}{38} + y = 66$

Subtract 38 from both sides

$\textcolor{red}{- 38} + 38 + y = 66 \textcolor{red}{- 38}$

$y = 28$

$38$ questions are valued at 7 points each.
$28$ questions are valued at 4 points each.

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 $y$.

$x + y = 66$

Subtract x from both sides

$\textcolor{red}{- x +} x + y = 66 \textcolor{red}{- x}$

y=66-x#

Now substitute this into the second equation

$7 x + 4 \left(\textcolor{red}{66 - x}\right) = 378$

Distribute

$7 x + \left(\textcolor{red}{264 - 4 x}\right) = 378$

Combine like terms

$3 x + 264 = 378$

Subtract 264 from both sides

$3 x + 264 \textcolor{red}{- 264} = 378 \textcolor{red}{- 264}$

Simplify

$3 x = 114$

Isolate $x$ by dividing by $3$

$\cancel{3} \frac{x}{\cancel{3}} = \frac{114}{3}$

$x = 38$

Now substitute this value for $x$ in the original version of the first equation.

$\textcolor{red}{38} + y = 66$

Subtract 38 from both sides

$\textcolor{red}{- 38} + 38 + y = 66 \textcolor{red}{- 38}$

$y = 28$

$38$ questions are valued at 7 points each.
$28$ questions are valued at 4 points each.

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