A carpenter purchased 50 ft of redwood and 70 ft of pine for a total cost of $258. A second purchase, at the same prices, included 90 ft of redwood and 60 ft of pine for a total cost of $372. Find the cost per foot of redwood and of pine.

Step-by-step explanation:

The two purchases can be described by the equations ...

  50r +70p = 258

  90r +60p = 372

Rewriting these equations in general form facilitates the use of the cross-multiplication method of solving them.

  50r +70p -258 = 0

  90r +60p -372 = 0

According to the cross-multiplication method, we need three cross-products:

  ∆1 = (50)(60) -(90)(70) = -3300

  ∆2 = (70)(-372) -(60)(-258) = -10560

  ∆3 = (-258)(90) -(-372)(50) = -4620

The solutions are the solutions to the equations ...

  1/∆1 = t/∆2 = p/∆3

  r = ∆2/∆1 = -10560/-3300 = 3.20

  p = ∆3/∆1 = -4620/-3300 = 1.40

The cost per foot of the redwood was $3.20; of the pine, $1.40.

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Additional comment

Here's how this version of the "cross-multiplication" method works. For equations ...

a coefficient array can be written as ...

The cross-products of interest are formed from adjacent columns:

and the solutions are ...

Using this method requires no more arithmetic operations than solving by substitution or elimination, and may require fewer: 6 products, 3 sums, and 2 quotients are needed.

51-50

step-by-step explanation:

d, x=8

step-by-step explanation:

simplifying

4x + -5 = 2x + 11

reorder the terms:

-5 + 4x = 2x + 11

reorder the terms:

-5 + 4x = 11 + 2x

solving

-5 + 4x = 11 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-5 + 4x + -2x = 11 + 2x + -2x

combine like terms: 4x + -2x = 2x

-5 + 2x = 11 + 2x + -2x

combine like terms: 2x + -2x = 0

-5 + 2x = 11 + 0

-5 + 2x = 11

add '5' to each side of the equation.

-5 + 5 + 2x = 11 + 5

combine like terms: -5 + 5 = 0

0 + 2x = 11 + 5

2x = 11 + 5

combine like terms: 11 + 5 = 16

2x = 16

divide each side by '2'.