Question

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Explanation:

43a)

Step1: Calculate Slope

Using two points \((0, 6)\) and \((1, 7)\), slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{7 - 6}{1 - 0}=1\).

Step2: Find y - intercept

When \(x = 0\), \(y = 6\), so y - intercept \(b = 6\).

Step3: Write Equation

Slope - intercept form is \(y=mx + b\), so \(y=x + 6\).

Step1: Calculate Slope

Using points \((-2, 6)\) and \((-4, 10)\), \(m=\frac{10 - 6}{-4-(-2)}=\frac{4}{-2}=-2\).

Step2: Find y - intercept

Use point \((-2, 6)\) and \(m=-2\) in \(y=mx + b\).
\(6=-2\times(-2)+b\Rightarrow6 = 4 + b\Rightarrow b = 2\).

Step3: Write Equation

\(y=-2x + 2\).

Step1: Substitute into Slope Formula

Given \(m=-4\), \((x_1,y_1)=(12,10)\), \((x_2,y_2)=(-2,r)\).
\(-4=\frac{r - 10}{-2-12}\).

Step2: Solve for r

Simplify denominator: \(-2-12=-14\).
\(-4=\frac{r - 10}{-14}\).
Multiply both sides by \(-14\): \(-4\times(-14)=r - 10\Rightarrow56=r - 10\).
Add 10 to both sides: \(r=56 + 10 = 66\).

Answer:

slope: \(1\)
y - intercept: \(6\)
equation: \(y=x + 6\)

43b)