Question

solving by substitution
mr. martin’s math test, which is worth 100 points, has 29 problems. each problem is worth either 5 points or 2 points.
let ( x ) be the number of questions worth 5 points and let ( y ) be the number of questions worth 2 points.
( x + y = 29 ), ( 5x + 2y = 100 )
how many problems of each point value are on the test?

• 10 problems worth 5 points and 25 problems worth 2 points
• 14 problems worth 5 points and 15 problems worth 2 points
• 16 problems worth 5 points and 13 problems worth 2 points
• 15 problems worth 5 points and 13 problems worth 2 points

Explanation:

Step1: Isolate $x$ from first equation

From $x + y = 29$, we get $x = 29 - y$.

Step2: Substitute into second equation

Substitute $x=29-y$ into $5x + 2y = 100$:
$$5(29 - y) + 2y = 100$$
Expand and simplify:
$$145 - 5y + 2y = 100$$
$$145 - 3y = 100$$

Step3: Solve for $y$

Rearrange to solve for $y$:
$$-3y = 100 - 145$$
$$-3y = -45$$
$$y = \frac{-45}{-3} = 15$$

Step4: Solve for $x$

Substitute $y=15$ back into $x=29-y$:
$$x = 29 - 15 = 14$$

Answer:

14 problems worth 5 points and 15 problems worth 2 points