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

Explanation:

Step1: Recall Form Definitions

- **Point - slope form**: \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point and \(m\) is the slope.
- **Slope - intercept form**: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
- **Standard form**: \(Ax+By = C\), where \(A\), \(B\), and \(C\) are integers and \(A\geq0\).

Step2: Analyze Each Equation

- **For \(y=\frac{2}{3}x + 17\)**:
- It is in the form \(y=mx + b\) (where \(m = \frac{2}{3}\) and \(b = 17\)), so it is in slope - intercept form.
- **For \(y - 7=\frac{2}{3}(x + 15)\)**:
- It is in the form \(y - y_1=m(x - x_1)\) (where \(x_1=- 15\), \(y_1 = 7\), and \(m=\frac{2}{3}\)), so it is in point - slope form.
- **For \(-\frac{2}{3}x + y=17\)**:
- Rearranging gives \(y=\frac{2}{3}x + 17\) (slope - intercept), but in its given form \(Ax+By = C\) (where \(A=-\frac{2}{3}\), \(B = 1\), \(C = 17\)). However, if we multiply through by 3 to get integer coefficients: \(- 2x+3y = 51\) or \(2x-3y=-51\) (standard form with \(A = 2\), \(B=-3\), \(C=-51\)). But the original form \(-\frac{2}{3}x + y=17\) can also be seen as a non - standard standard form. Wait, actually, the equation \(6x - 3y=-51\) can be simplified. Divide \(6x - 3y=-51\) by 3: \(2x - y=-17\) or multiply by - 1: \(-2x + y = 17\), which is equivalent to \(-\frac{2}{3}x + y=17\) when we consider the slope - intercept form derivation. Wait, let's re - evaluate. The standard form is \(Ax + By=C\) with \(A\), \(B\), \(C\) integers and \(A\geq0\). For \(6x-3y=-51\), we can divide by 3: \(2x - y=-17\) (standard form). For \(-\frac{2}{3}x + y=17\), multiply by 3: \(-2x + 3y = 51\) or \(2x-3y=-51\), which is the same as \(6x - 3y=-51\) when we multiply \(2x-3y=-51\) by 3? No, \(2x-3y=-51\) multiplied by 3 is \(6x - 9y=-153\). Wait, I made a mistake. Let's do it correctly. \(6x-3y=-51\) divide by 3: \(2x - y=-17\). \(-\frac{2}{3}x + y=17\) multiply by 3: \(-2x + 3y = 51\) or \(2x-3y=-51\). Ah, so \(6x - 3y=-51\) can be simplified by dividing by 3: \(2x - y=-17\), but \(-\frac{2}{3}x + y=17\) is equivalent to \(2x-3y=-51\) (multiply both sides by - 3: \(2x-3y=-51\)). So \(6x - 3y=-51\) is also a standard form (since \(A = 6\), \(B=-3\), \(C=-51\), and we can simplify to \(2x - y=-17\) by dividing by 3).
- **For \(x - 7=\frac{2}{3}(y + 15)\)**: This is a linear equation in terms of \(x\) and \(y\) but is not in the point - slope form of \(y\) in terms of \(x\) (the point - slope form for \(y\) is \(y - y_1=m(x - x_1)\)), so it is not in the point - slope form of the \(y\) - based linear equation we are considering (since point - slope is for \(y\) as a function of \(x\) in the form \(y - y_1=m(x - x_1)\)).

Step3: Match the Forms

- **Point - slope form**: The equation \(y - 7=\frac{2}{3}(x + 15)\) matches the point - slope form \(y - y_1=m(x - x_1)\) (here \(x_1=-15\), \(y_1 = 7\), \(m=\frac{2}{3}\)).
- **Slope - intercept form**: The equation \(y=\frac{2}{3}x + 17\) matches the slope - intercept form \(y = mx + b\) (here \(m=\frac{2}{3}\), \(b = 17\)).
- **Standard form**: The equation \(6x - 3y=-51\) (or \(-\frac{2}{3}x + y=17\) which is equivalent to \(2x-3y=-51\) and \(6x - 3y=-51\) can be seen as a standard form with \(A = 6\), \(B=-3\), \(C=-51\)) is in standard form \(Ax + By=C\). Also, \(-\frac{2}{3}x + y=17\) is a linear equation in standard form (even with a fraction for \(A\), but usually standard form has integer coefficients, so \(6x - 3y=-51\) is a better candidate as it has integer coefficients.

Step4: Match the Pairs

- **Point - slope form**: \(y - 7=\frac{2}{3}(x + 15)\)
- **Slope - intercept form**: \(y=\frac{2}{3}x + 17\)
- **Standard form**: \(6x - 3y=-51\) (or \(-\frac{2}{3}x + y=17\), but \(6x - 3y=-51\) has integer coefficients)

Answer:

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