Question

1. which number is the solution to \\(\frac{2}{5}x + 9 = x - 3\\)?

a. 10
b. 15
c. 20
d. 25

1. consider \\(x + 3 = 7\\). how do you know that 4 is the solution to this equation?

1. match each situation with the equation that best represents the situation.

situation

a. there are 8 nails in a toolbox. the number of rubber bands in the toolbox is 24 more than the number of nails. let \\(y\\) represent the number of rubber bands in the toolbox.

b. there are 24 small dogs at a park. there are 8 more small dogs than large dogs at the park. let \\(y\\) represent the number of large dogs at the park.

c. there are 8 pizzas served at a party. 24 people share the pizzas equally. let \\(y\\) represent the fraction of pizza each person receives.

d. the area of a rectangle is 24 square feet. the length of the rectangle is 8 feet. let \\(y\\) represent the width of the rectangle in feet.

equation
\\(24 = y + 8\\)
\\(24 = 8y\\)
\\(8 = 24y\\)
\\(24 = y - 8\\)

Explanation:

Step1: Isolate x terms

$\frac{2}{5}x + 9 = x - 3$
Subtract $\frac{2}{5}x$ from both sides:
$9 = x - \frac{2}{5}x - 3$
$9 = \frac{3}{5}x - 3$

Step2: Isolate the x term

Add 3 to both sides:
$9 + 3 = \frac{3}{5}x$
$12 = \frac{3}{5}x$

Step3: Solve for x

Multiply both sides by $\frac{5}{3}$:
$x = 12 \times \frac{5}{3}$
$x = 20$
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Step4: Verify solution for $x+3=7$

Substitute $x=4$ into the left side:
$4 + 3 = 7$

Step5: Compare to right side

The left side equals the right side ($7=7$), so 4 is a solution.
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Step6: Match situation a to equation

Situation a: Rubber bands ($y$) are 24 more than 8 nails: $y = 8 + 24$ → $24 = y - 8$

Step7: Match situation b to equation

Situation b: 24 small dogs are 8 more than large dogs ($y$): $24 = y + 8$

Step8: Match situation c to equation

Situation c: 8 pizzas split by 24 people: $8 = 24y$

Step9: Match situation d to equation

Situation d: Area = length × width: $24 = 8y$

Answer:

1. C. 20
2. Substitute $x=4$ into $x+3=7$: $4+3=7$, which equals the right-hand side, so 4 is the solution.

12.
a. $24 = y - 8$
b. $24 = y + 8$
c. $8 = 24y$
d. $24 = 8y$