Question

we want to write an equation for the line, shown below, that passes through the origin (0, 0) and the point (x, y).
lets start with the equation for the slope of a line, m, through any two points ((x_1, y_1)) and ((x_2, y_2)). (\frac{y_2 - y_1}{x_2 - x_1} = m)
step 1: complete the substitution, using (0, 0) for ((x_1, y_1)) and (x, y) for ((x_2, y_2)) in the above equation. (\frac{y - square}{square - square} = m)
step 2: simplify the numerator and the denominator. (\frac{square}{square} = m)
step 3: solve for y. (y = square)

Explanation:

Step 1: Substitute the points

We know that \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (x, y)\). Substituting into the slope formula \(\frac{y_2 - y_1}{x_2 - x_1}=m\), we get \(\frac{y - 0}{x - 0}=m\).

Step 2: Simplify numerator and denominator

Simplifying the numerator \(y - 0 = y\) and the denominator \(x - 0 = x\), so we have \(\frac{y}{x}=m\) (assuming \(x
eq0\)).

Step 3: Solve for \(y\)

Multiply both sides of the equation \(\frac{y}{x}=m\) by \(x\) (for \(x
eq0\)) to get \(y = mx\). Since the line passes through the origin, this equation also holds for \(x = 0\) (as \(y = 0\) when \(x = 0\) and \(y=mx\) gives \(y = 0\) when \(x = 0\)).

Filling in the blanks:

• Step 1: \(\frac{y - \boldsymbol{0}}{x - \boldsymbol{0}}=m\)
• Step 2: \(\frac{\boldsymbol{y}}{\boldsymbol{x}}=m\)
• Step 3: \(y=\boldsymbol{mx}\)

Answer:

• Step 1: Numerator blank: \(0\), Denominator blanks: \(x\), \(0\) (so \(\frac{y - 0}{x - 0}=m\))
• Step 2: Numerator: \(y\), Denominator: \(x\) (so \(\frac{y}{x}=m\))
• Step 3: \(y = mx\) (the blank is filled with \(mx\))