Question

which of the following represent the same line? select all that apply.\\(\square\\) \\(y = 3 - 2x\\)\\(\square\\) \\(f(x) = -2x + 3\\)\\(\square\\) \\(f(x) = 2x - 3\\)\\(\square\\) \\(y = -2x + 3\\)\
without plotting the points, explain how you can determine whether the three points \\((-1, -3)\\), \\((2, 1)\\), and \\((8, 9)\\) lie on the same line, and then click check to compare your answer to a sample answer.\
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write the equation of the line in slope - intercept form that is perpendicular to \\(y = \frac{4}{3}x - 3\\), and passes through \\((-2, 5)\\).\
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Explanation:

First Sub - Question (Which represent the same line)

To determine which equations represent the same line, we can rewrite them in slope - intercept form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept) and check for equal slopes and y - intercepts.

• For \(y = 3-2x\), we can rewrite it as \(y=-2x + 3\) (by rearranging the terms).
• For \(f(x)=-2x + 3\), since \(y = f(x)\), this is already in slope - intercept form with \(m=-2\) and \(b = 3\).
• For \(f(x)=2x-3\), the slope \(m = 2\) and y - intercept \(b=-3\), which is different from the others.
• For \(y=-2x + 3\), it has slope \(m=-2\) and y - intercept \(b = 3\), same as the rewritten \(y = 3-2x\) and \(f(x)=-2x + 3\).

To determine if three points \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\) lie on the same line, we can calculate the slope between the first and second point, and then the slope between the second and third point. If the two slopes are equal, the points lie on the same line. The formula for slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

• Slope between \((-1,-3)\) and \((2,1)\): \(m_1=\frac{1-(-3)}{2-(-1)}=\frac{4}{3}\)
• Slope between \((2,1)\) and \((8,9)\): \(m_2=\frac{9 - 1}{8 - 2}=\frac{8}{6}=\frac{4}{3}\)

Since \(m_1=m_2\), the three points lie on the same line.

Step 1: Find the slope of the perpendicular line

The slope of the given line \(y=\frac{4}{3}x-3\) is \(m_1=\frac{4}{3}\). The slope of a line perpendicular to a line with slope \(m_1\) is \(m_2=-\frac{1}{m_1}\) (negative reciprocal). So \(m_2=-\frac{3}{4}\).

Step 2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-2,5)\) and \(m =-\frac{3}{4}\).
Substitute the values: \(y - 5=-\frac{3}{4}(x+2)\)

Step 3: Convert to slope - intercept form

Expand the right - hand side: \(y - 5=-\frac{3}{4}x-\frac{3}{2}\)
Add 5 to both sides: \(y=-\frac{3}{4}x-\frac{3}{2}+5\)
Simplify the constant terms: \(5=\frac{10}{2}\), so \(y=-\frac{3}{4}x+\frac{10 - 3}{2}=-\frac{3}{4}x+\frac{7}{2}\)

Answer:

\(y = 3-2x\), \(f(x)=-2x + 3\), \(y=-2x + 3\)

Second Sub - Question (Three points on the same line)