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Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (-3,5) and parallel to x + 4y=7

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Answer:

  • [tex]\boxed{\sf Standard-form :x + 4y -17=0 }\\[/tex]

  • [tex]\boxed{\sf Slope-intercept\ form :y =\dfrac{-1}{4}x +\dfrac{17}{4}}[/tex]

Step-by-step explanation:

Here a equation of the line is given to us and we need to find out the equation of line which passes through the given point and parallel to the given line , the given equation is ,

[tex]\longrightarrow x + 4y = 7\\[/tex]

Firstly convert it into slope intercept form of the line which is y = mx + x , as ;

[tex]\longrightarrow 4y = -x + 7 \\[/tex]

[tex]\longrightarrow y =\dfrac{-x}{4}+\dfrac{7}{4}\\[/tex]

On comparing it to y = mx + c , we have ,

[tex]\longrightarrow m =\dfrac{-1}{4}\\[/tex]

[tex]\longrightarrow c =\dfrac{7}{4}\\[/tex]

Now as we know that the slope of two parallel lines is same . Therefore the slope of the parallel line will be ,

[tex]\longrightarrow m_{||)}=\dfrac{-1}{4}\\[/tex]

Now we may use point slope form of the line as ,

[tex]\longrightarrow y - y_1 = m(x-x_1) \\[/tex]

On substituting the respective values ,

[tex]\longrightarrow y - 5 =\dfrac{-1}{4}\{ x -(-3)\}\\[/tex]

[tex]\longrightarrow y -5=\dfrac{-1}{4}(x+3)\\[/tex]

[tex]\longrightarrow 4(y -5 ) =-1(x +3) \\[/tex]

[tex]\longrightarrow 4y -20 = - x -3 \\[/tex]

[tex]\longrightarrow x + 4y -20+3=0\\[/tex]

[tex]\longrightarrow \underset{Standard \ Form }{\underbrace{\underline{\underline{ x + 4y -17=0}}}} \\[/tex]

Again the equation can be rewritten as ,

[tex]\longrightarrow y - 5 = \dfrac{-1}{4}(x +3) \\[/tex]

[tex]\longrightarrow y = \dfrac{-1}{4}x -\dfrac{3}{4}+5 \\[/tex]

[tex]\longrightarrow y = \dfrac{-1}{4}x -\dfrac{20-3}{4} \\[/tex]

[tex]\longrightarrow \underset{Slope-Intercept\ form }{\underbrace{\underline{\underline{ y =\dfrac{-1}{4}x +\dfrac{17}{4}}}}}\\[/tex]

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