LINEAR EQUATIONS $\newline$ WARM-UP $\newline$ $1$ . Kia's plant is $8 \frac{1}{5}$ inches tall and is growing $\frac{2}{3}$ inch each week. Lyric's plant is $10 \frac{7}{10}$ inches tall and is growing $\frac{7}{6}$ inch each week. Write and solve an equation to find how many weeks it will take for the height of the two plants to be the same. $\newline$ Equation: $\qquad$ $\newline$ Solution: $\qquad$

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Q. LINEAR EQUATIONS

\newline

WARM-UP

\newline

1

. Kia's plant is

8 \frac{1}{5}

inches tall and is growing

\frac{2}{3}

inch each week. Lyric's plant is

10 \frac{7}{10}

inches tall and is growing

\frac{7}{6}

inch each week. Write and solve an equation to find how many weeks it will take for the height of the two plants to be the same.

\newline

Equation:

\qquad

\newline

Solution:

\qquad

Convert to Improper Fraction: Kia's plant height after $w$ weeks: $8\left(\frac{1}{5}\right) + \left(\frac{2}{3}\right)w$ inches. $\newline$ Convert mixed number to improper fraction: $8\left(\frac{1}{5}\right) = \frac{41}{5}$ . $\newline$ Kia's plant height equation: $\left(\frac{41}{5}\right) + \left(\frac{2}{3}\right)w$ .
Kia's Plant Height Equation: Lyric's plant height after $w$ weeks: $10\left(\frac{7}{10}\right) + \left(\frac{7}{6}\right)w$ inches. $\newline$ Convert mixed number to improper fraction: $10\left(\frac{7}{10}\right) = \frac{107}{10}$ . $\newline$ Lyric's plant height equation: $\left(\frac{107}{10}\right) + \left(\frac{7}{6}\right)w$ .
Lyric's Plant Height Equation: Set the two plant heights equal to find when they will be the same: $\newline$ $(\frac{41}{5}) + (\frac{2}{3})w = (\frac{107}{10}) + (\frac{7}{6})w$ .
Set Equal and Solve: To solve for $w$ , first get all $w$ terms on one side and constants on the other: $\frac{2}{3}w - \frac{7}{6}w = \frac{107}{10} - \frac{41}{5}$ .
Combine W Terms: Find a common denominator for the w terms: $6$ . $\newline$ $(\frac{4}{6})w - (\frac{7}{6})w = (\frac{107}{10}) - (\frac{41}{5})$ . $\newline$ Combine w terms: $-(\frac{3}{6})w = (\frac{107}{10}) - (\frac{41}{5})$ .
Simplify Right Side: Simplify the right side by finding a common denominator: $10$ .
$-(\frac{3}{6})w = (\frac{107}{10}) - (\frac{82}{10})$ .
Combine constants: $-(\frac{3}{6})w = \frac{25}{10}$ .
Simplify Both Sides: Simplify both sides: $-(\frac{1}{2})w = \frac{5}{2}$ . Multiply both sides by $-2$ to solve for $w$ : $w = -5$ .