Question

rewriting literal equations
example 3 solve the literal equation -5y - 2x = 10 for y.
-5y - 2x = 10
-5y - 2x + 2x = 10 + 2x
-5y = 10 + 2x
\frac{-5y}{-5} = \frac{10 + 2x}{-5}
y = -2 - \frac{2}{5}x

solve the literal equation for y.

1. 4x + y = 2
2. x - \frac{1}{3}y = -1
3. 2y - 9 = 13x
4. 2xy + 6y = 10
5. 8x - 4xy = 3
6. 6x + 7xy = 15

1. discuss mathematical thinking is the order in which you apply properties of exponents important?

explain your reasoning.

Explanation:

7. Step1: Isolate $y$

$4x + y = 2$
$y = 2 - 4x$

8. Step1: Isolate $y$-term

$x - \frac{1}{4}y = -1$
$-\frac{1}{4}y = -1 - x$

8. Step2: Multiply by -4

$y = (-1 - x) \times (-4)$
$y = 4 + 4x$

9. Step1: Isolate $2y$

$2y - 9 = 13x$
$2y = 13x + 9$

9. Step2: Divide by 2

$y = \frac{13x + 9}{2}$
$y = \frac{13}{2}x + \frac{9}{2}$

10. Step1: Factor out $y$

$2xy + 6y = 10$
$y(2x + 6) = 10$

10. Step2: Divide by $2x+6$

$y = \frac{10}{2x + 6}$
$y = \frac{5}{x + 3}$

11. Step1: Factor out $x$

$8x - 4xy = 3$
$x(8 - 4y) = 3$

11. Step2: Isolate $y$-term

$8 - 4y = \frac{3}{x}$
$-4y = \frac{3}{x} - 8$

11. Step3: Divide by -4

$y = \frac{8 - \frac{3}{x}}{4}$
$y = 2 - \frac{3}{4x}$

12. Step1: Isolate $y$-term

$6x + 7xy = 15$
$7xy = 15 - 6x$

12. Step2: Divide by $7x$

$y = \frac{15 - 6x}{7x}$
$y = \frac{15}{7x} - \frac{6}{7}$

13. Step1: Explain exponent order importance

Exponent properties (e.g., $(a^m)^n=a^{mn}$, $a^m \cdot a^n=a^{m+n}$) have strict rules. Swapping order can lead to different results. For example, $(2^3)^2=2^6=64$, but $2^{(3^2)}=2^9=512$.

Answer:

1. $y = 2 - 4x$
2. $y = 4 + 4x$
3. $y = \frac{13}{2}x + \frac{9}{2}$
4. $y = \frac{5}{x + 3}$
5. $y = 2 - \frac{3}{4x}$
6. $y = \frac{15}{7x} - \frac{6}{7}$
7. Yes, the order of applying exponent properties is important. Different orders can produce different results, as exponent properties follow specific hierarchical rules (e.g., power of a power vs. exponentiation of a base to a power of a power yield distinct values).