Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Use the following scenario for 1 - 3:

One gram of protein contains 4 calories. One gram of fat contains 9 calories. A snack has 60 calories from $p$ grams of protein and $f$ grams of fat.

1.

Determine if 5 grams of protein and 2 grams of fat could be the number of grams of protein and fat in the snack. Explain your reasoning.

Compare your answer:

No, because $4 (5) + 9 (2) = 20 + 18 = 38$ , not 60.

2.

Determine if 10.5 grams of protein and 2 grams of fat could be the number of grams of protein and fat in the snack. Explain your reasoning.

Compare your answer:

Yes, because $4 (10.5) + 9 (2) = 42 + 18 = 60$ .

3.

Determine if 8 grams of protein and 4 grams of fat could be the number of grams of protein and fat in the snack. Explain your reasoning.

Compare your answer:

No, because $4 (8) + 9 (4) = 32 + 36 = 68$ , not 60.

4.

If there are 6 grams of fat in the snack, how many grams of protein are there? Be prepared to show your reasoning.

Compare your answer:

1.5 grams. For example: $4 p + 9 (6) = 60$ , so $p$ must be 1.5 for the equation to be true.

5.

In this situation, what does a solution to the equation $4 p + 9 f = 60$ tell us? Give an example of a solution.

Compare your answer:

It means a pair of grams of protein and fat in the snack that add up to 60 calories. One example would be 6 grams of protein and 4 grams of fat.

Video: Working Through the Equation

Watch the following video to learn more about how to determine a solution to this particular equation: $4 p + 9 f = 60$ .

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Self Check

Which of the following is a solution (

x, y

) to

2x + 3y = 7

?

$(-2, 4)$
$(4, -1)$
$(1, 2)$
$(2, 1)$

Additional Resources

Solutions to Equations in Two Variables

An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables.

Drawing of equations of two variables. f equals nine fifths c plus 32 to convert temperature from degrees fahrenheit F to degrees celsisus C and a formula i equals two point five four c and coverts a measure from cenimenters c to inches i.

A solution to such an equation is a pair of numbers that makes the equation true.

Example

Suppose Tyler spends $45 on T-shirts and socks. A T-shirt costs $10, and a pair of socks costs $2.50. If $t$ represents the number of T-shirts and $p$ represents the number of pairs of socks that Tyler buys, we can represent this situation with the equation:

$10 t + 2.50 p = 45$

This is an equation in two variables. More than one pair of values for $t$ and $p$ make the equation true.

Which pair of values makes the equation $10 t + 2.50 p = 45$ true?

1.

$t = 3$ and $p = 6$

These values make the equation true, because:

$10 (3) + 2.50 (6) = 45$

$30 + 15 = 45$

$45 = 45$

2.

$t = 4$ and $p = 2$

These values make the equation true, because:

$10 (4) + 2.50 (2) = 45$

$40 + 5 = 45$

$45 = 45$

3.

$t = 1$ and $p = 10$

These values make the equation false, because:

$10 (1) + 2.50 (10) = 45$

$10 + 25 = 45$

$35 \neq 45$

In this situation, one constraint is that the combined cost of shirts and socks must equal $45.

Solutions to the equations are pairs of $t$ and $p$ values that satisfy this constraint, such as in questions 1 - 2.

Combinations such as $t = 1$ and $p = 10$ , as in question 3, are not solutions because they don't meet the constraint. When these pairs of values are substituted into the equation, they result in statements that are false.

Try it

Try It: Solutions to Equations in Two Variables

Is $a = 3$ and $b = 5$ a solution to $6 a - 3 b = 3$ ?

These values make the equation true.

They are a solution to the equation, because:

$6 (3) - 3 (5) = 3$

$18 - 15 = 3$

$3 = 3$

1.4.3 Finding the Solution to an Equation in Two Variables

Activity

Video: Working Through the Equation

Additional Resources

Solutions to Equations in Two Variables

Try It: Solutions to Equations in Two Variables