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$12t = 4v - 3$ $\newline$ $-6t = 4v + 6$ $\newline$ If $(t, v)$ is the solution to the system of equations, what is the value of $t - v$ ?

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Write and solve direct variation equations

Full solution

Q.

12t = 4v - 3

\newline

-6t = 4v + 6

\newline

If

(t, v)

is the solution to the system of equations, what is the value of

t - v

?

Write System Equations: Write down the system of equations. $\newline$ We have the following system of equations: $\newline$ $1$ ) $12t = 4v - 3$ $\newline$ $2$ ) $-6t = 4v + 6$
Solve First Equation for v: Solve the first equation for v. $\newline$ From the first equation, we can express $v$ in terms of $t$ : $\newline$ $12t = 4v - 3$ $\newline$ Add $3$ to both sides: $\newline$ $12t + 3 = 4v$ $\newline$ Divide both sides by $4$ : $\newline$ $(12t + 3)/4 = v$ $\newline$ $v = 3t + 3/4$
Substitute $v$ into Second Equation: Substitute the expression for $v$ from Step $2$ into the second equation. $\newline$ We have $v = 3t + \frac{3}{4}$ , so we substitute this into the second equation: $\newline$ \(-6t = $4$ ( $3$ t + \frac{ $3$ }{ $4$ }) + $6$
Distribute and Simplify: Distribute and simplify the second equation. $\newline$ $-6t = 4(3t) + 4(\frac{3}{4}) + 6$ $\newline$ $-6t = 12t + 3 + 6$ $\newline$ Combine like terms: $\newline$ $-6t = 12t + 9$
Solve for t: Solve for t. $\newline$ Subtract $12t$ from both sides: $\newline$ $-6t - 12t = 9$ $\newline$ $-18t = 9$ $\newline$ Divide both sides by $-18$ : $\newline$ $t = \frac{9}{-18}$ $\newline$ $t = -\frac{1}{2}$
Substitute $t$ into $v$ Expression: Substitute $t$ back into the expression for $v$ from Step $2$ . $\newline$ $v = 3t + \frac{3}{4}$ $\newline$ $v = 3(-\frac{1}{2}) + \frac{3}{4}$ $\newline$ $v = -\frac{3}{2} + \frac{3}{4}$
Combine Terms for $v$ : Find a common denominator and combine the terms to solve for $v$ . $v = -\frac{6}{4} + \frac{3}{4}$ $v = \frac{(-6 + 3)}{4}$ $v = -\frac{3}{4}$
Calculate $t - v$ : Calculate $t - v$ .
$t - v = (-\frac{1}{2}) - (-\frac{3}{4})$
Find a common denominator:
$t - v = (-\frac{2}{4}) - (-\frac{3}{4})$
$t - v = -\frac{2}{4} + \frac{3}{4}$
$t - v = \frac{1}{4}$

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