Question

question
drag the tiles to the correct boxes to complete the pairs. not all tiles will be used.
match each linear equation with the correct description.
tiles
$\boldsymbol{-\frac{2}{3}x + y = 17}$ $\boldsymbol{6x - 3y = -51}$ $\boldsymbol{y - 7 = \frac{2}{3}(x + 15)}$ $\boldsymbol{y = \frac{2}{3}x + 17}$
$\boldsymbol{x - 7 = \frac{2}{3}(y + 15)}$
pairs
standard form
slope - intercept form
point - slope form

Explanation:

Step1: Recall Form Definitions

• Standard Form: \(Ax + By = C\) (A, B, C integers, A≥0)
• Slope - Intercept Form: \(y = mx + b\) (m = slope, b = y - intercept)
• Point - Slope Form: \(y - y_1 = m(x - x_1)\) (m = slope, \((x_1,y_1)\) = point)

Step2: Analyze Each Equation

Standard Form Candidates:

• For \(-\frac{2}{3}x + y = 17\): Multiply by 3 to get \(-2x + 3y = 51\) (or \(2x - 3y=-51\)), but original is \(Ax + By = C\) (A = \(-\frac{2}{3}\), B = 1, C = 17). Also, \(6x - 3y=-51\) is in \(Ax + By = C\) (A = 6, B = - 3, C=-51).

Slope - Intercept Form Candidates:

• \(y=\frac{2}{3}x + 17\) is \(y = mx + b\) (m=\(\frac{2}{3}\), b = 17). Also, solve \(-\frac{2}{3}x + y = 17\) for y: \(y=\frac{2}{3}x + 17\), so it can be rewritten to slope - intercept.

Point - Slope Form Candidates:

• \(y - 7=\frac{2}{3}(x + 15)\) is \(y - y_1 = m(x - x_1)\) ( \(y_1 = 7\), \(x_1=-15\), m=\(\frac{2}{3}\)).

Step3: Match Each Form

• Standard Form: \(6x - 3y=-51\) (fits \(Ax + By = C\) with A = 6, B=-3, C = - 51) and \(-\frac{2}{3}x + y = 17\) (fits \(Ax + By = C\) with A=\(-\frac{2}{3}\), B = 1, C = 17). But \(6x - 3y=-51\) is a cleaner standard form with integer coefficients.
• Slope - Intercept Form: \(y=\frac{2}{3}x + 17\) (directly in \(y = mx + b\)) and \(-\frac{2}{3}x + y = 17\) (after solving for y: \(y=\frac{2}{3}x + 17\)).
• Point - Slope Form: \(y - 7=\frac{2}{3}(x + 15)\) (matches \(y - y_1 = m(x - x_1)\) with \(x_1=-15\), \(y_1 = 7\), m=\(\frac{2}{3}\)).

Step4: Final Matching

• Standard Form: \(6x - 3y=-51\) (or \(-\frac{2}{3}x + y = 17\))
• Slope - Intercept Form: \(y=\frac{2}{3}x + 17\) (or \(-\frac{2}{3}x + y = 17\) when solved for y)
• Point - Slope Form: \(y - 7=\frac{2}{3}(x + 15)\)

Answer:

• standard form: \(6x - 3y=-51\) (and \(-\frac{2}{3}x + y = 17\))
• slope - intercept form: \(y=\frac{2}{3}x + 17\) (and \(-\frac{2}{3}x + y = 17\))
• point - slope form: \(y - 7=\frac{2}{3}(x + 15)\)

(Note: For the drag - and - drop, the most direct matches are:

• standard form: \(6x - 3y=-51\)
• slope - intercept form: \(y=\frac{2}{3}x + 17\)
• point - slope form: \(y - 7=\frac{2}{3}(x + 15)\)

)