Write an equation in point-slope form and slope-intercept form of the line passing through (-10,3) and (-2,-5).

Verified step by step guidance

1. First, find the slope (m) of the line passing through the points (-10, 3) and (-2, -5) using the slope formula: $m=\frac{y\_2\; -y\_1}{x\_2\; -x\_1}$. Substitute the values: $m\; =\frac{-5\; -\; 3}{-2\; -\; (-10)}$.
2. Simplify the numerator and denominator to find the slope: $m\; =\frac{-8}{8}$. Then simplify the fraction to get the slope value.
3. Use the point-slope form of a line equation: $y\; -\; y\_1\; =\; m(x\; -\; x\_1)$. Choose one of the given points, for example (-10, 3), and substitute $m$, $x\_1\; =\; -10$, and $y\_1\; =\; 3$ into the equation.
4. Write the point-slope form equation explicitly with the substituted values: $y\; -\; 3\; =\; m(x\; +\; 10)$.
5. To convert to slope-intercept form, solve the point-slope equation for $y$ by distributing the slope and isolating $y$ on one side, resulting in an equation of the form $y\; =\; mx\; +\; b$.

Key Concepts

Slope of a Line

The slope measures the steepness of a line and is calculated as the change in y-values divided by the change in x-values between two points. For points (x₁, y₁) and (x₂, y₂), slope m = (y₂ - y₁) / (x₂ - x₁). It indicates how much y changes for a unit change in x.

Point-Slope Form of a Line

Point-slope form expresses a line's equation using a known point (x₁, y₁) and the slope m: y - y₁ = m(x - x₁). This form is useful for writing the equation quickly when a point and slope are known, directly relating changes in x and y.

Slope-Intercept Form of a Line

Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. It clearly shows the slope and where the line crosses the y-axis, making it easy to graph and understand the line's behavior.