Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (−2, 2) and parallel to the line whose equation is 2x-3y-7=0

Accepted Answer

Identify the slope of the given line by rewriting its equation $$2x - 3y - 7 = 0$$ into slope-intercept form $$y = mx + b. $$Start by isolating $$y$$: add $$3y to $$both sides and subtract $$7$$ from both sides to get $$2x - 7 = 3y$$, then divide both sides by 3 to get $$y = \frac{2}{3}x - \frac{7}{3}. $$The slope $$m is $$therefore $$\frac{2}{3}. $$Since the new line is parallel to the given line, it will have the same slope $$m = \frac{2}{3}. $$Use the point-slope form of a line equation, which is $$y - y_1$ = m(x - x_1$)$$, where $$(x_1$, y_1$) is $$the point the line passes through. Substitute $$m = \frac{2}{3}$$ and the point $$(-2, 2)$$ into the formula. Write the point-slope form equation explicitly: $$y - 2 = \frac{2}{3}(x - (-2))$$, which simplifies to $$y - 2 = \frac{2}{3}(x + 2). To $$write the equation in general form, first eliminate the fraction by multiplying both sides of the equation by 3: $$3(y - 2) = 2(x + 2). $$Then expand both sides: $$3y - 6 = 2x + 4. $$Rearrange the equation to standard general form $$Ax + By + C = 0 by $$moving all terms to one side: $$2x - 3y + 10 = 0. $$This is the general form of the line passing through $$(-2, 2)$$ and parallel to the given line.

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (−2, 2) and parallel to the line whose equation is 2x-3y-7=0

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Key Concepts

Point-Slope Form of a Line

Parallel Lines and Their Slopes

General Form of a Line