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What is the equation in standard form of the line that passes through the point (6,-1) and is parallel to the line represented by 8x+3y=15 ?

What is the equation in standard form of the line that passes through the point (6,−1) (6,-1) and is parallel to the line represented by 8x+3y=15 8 x+3 y=15 ?

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Full solution

Q. What is the equation in standard form of the line that passes through the point (6,−1) (6,-1) and is parallel to the line represented by 8x+3y=15 8 x+3 y=15 ?
  1. Find Slope of Parallel Line: First, we need to find the slope of the line that is parallel to the given line. To do this, we will convert the equation 8x+3y=158x + 3y = 15 into slope-intercept form (y=mx+by = mx + b) to identify the slope.\newlineRearrange the equation to solve for y:\newline3y=−8x+153y = -8x + 15\newliney=(−8/3)x+5y = (-8/3)x + 5\newlineThe slope (mm) of the line is −8/3-8/3.
  2. Use Point-Slope Form: Since parallel lines have the same slope, the slope of the line we are looking for is also −83-\frac{8}{3}. Now we will use the point-slope form of the equation of a line, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is the point the line passes through.\newlineSubstitute the slope −83-\frac{8}{3} and the point (6,−1)(6, -1) into the point-slope form:\newliney−(−1)=(−83)(x−6)y - (-1) = \left(-\frac{8}{3}\right)(x - 6)
  3. Simplify Equation: Simplify the equation by distributing the slope on the right side and moving −1-1 to the other side:\newliney+1=(−83)x+(83)⋅6y + 1 = \left(-\frac{8}{3}\right)x + \left(\frac{8}{3}\right)\cdot6\newliney+1=(−83)x+16y + 1 = \left(-\frac{8}{3}\right)x + 16
  4. Convert to Standard Form: Now we need to convert this equation into standard form, which is Ax+By=CAx + By = C, where AA, BB, and CC are integers. To do this, we will first eliminate the fraction by multiplying every term by the denominator, which is 33: \newline3(y+1)=3(−83)x+3×163(y + 1) = 3\left(-\frac{8}{3}\right)x + 3\times 16
  5. Eliminate Fractions: Perform the multiplication: 3y+3=−8x+483y + 3 = -8x + 48
  6. Rearrange Terms: Next, we will move all terms involving variables to one side of the equation and the constant term to the other side to get the standard form: 8x+3y=48−38x + 3y = 48 - 3
  7. Find Constant Term: Subtract 33 from 4848 to find the constant term:\newline8x+3y=458x + 3y = 45\newlineThis is the equation of the line in standard form.

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