Write the equation of the line that passes through the points $(8,-1)$ and $(2,-5)$ in standard form, given slope form is $y+1=\frac{2}{3}(x-8)$.
$\square x+$ $\square y=$ $\square$
Answer (2)
The equation of the line in standard form that passes through the points (8, -1) and (2, -5) is 2x - 3y = 19. This was derived from the point-slope form provided. The steps included eliminating fractions, distributing, and rearranging to the standard form.
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Multiply both sides of the equation by 3: 3 ( y + 1 ) = 2 ( x − 8 ) .
Expand both sides: 3 y + 3 = 2 x − 16 .
Rearrange to standard form: 2 x − 3 y = 19 .
The equation in standard form is: 2 x − 3 y = 19 .
Explanation
Understanding the Problem We are given the equation of a line in point-slope form: y + 1 = 3 2 ( x − 8 ) . Our goal is to convert this equation into standard form, which is A x + B y = C , where A, B, and C are integers.
Eliminating the Fraction First, we want to eliminate the fraction by multiplying both sides of the equation by 3: 3 ( y + 1 ) = 3 × 3 2 ( x − 8 ) 3 ( y + 1 ) = 2 ( x − 8 )
Expanding the Equation Next, we distribute the constants on both sides of the equation: 3 y + 3 = 2 x − 16
Rearranging to Standard Form Now, we want to rearrange the equation to get it into the standard form A x + B y = C . We subtract 3 y from both sides and add 16 to both sides: 3 y + 3 − 3 y + 16 = 2 x − 16 − 3 y + 16 19 + 3 = 2 x − 3 y 2 x − 3 y = 19
Identifying Coefficients The equation is now in standard form: 2 x − 3 y = 19 . Therefore, A = 2 , B = − 3 , and C = 19 .
Final Answer Thus, the equation of the line in standard form is 2 x − 3 y = 19 .
Examples
Understanding the standard form of a line is crucial in various real-world applications. For instance, when designing a road, engineers use linear equations to model the slope and elevation changes. By converting the equation to standard form, they can easily identify key parameters such as the x and y intercepts, which represent critical points in the road's design. This ensures the road is safe and efficient for travel. Similarly, in construction, architects use linear equations to model the dimensions of buildings and ensure structural integrity.
