WEBVTT
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Write the equation of line that passes through the points two, negative two and negative two, 10 in the form ππ₯ plus ππ¦ plus π is equal to zero.
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So in this question, weβre asked to find the equation of a straight line given the coordinates of two points that lie on the line.
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Weβll answer this question using the slope-intercept method.
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First, weβll find the equation of the straight line in the form π¦ equals ππ₯ plus π, where π represents the slope of the line and π represents the π¦-intercept.
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This wonβt give the equation of the line in the specific form thatβs been asked for: ππ₯ plus ππ¦ plus π is equal to zero.
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So weβll need to do some rearrangement at the end.
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Letβs recall first how to calculate the slope of a line.
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The slope of the line joining the points with coordinates π₯ one, π¦ one and π₯ two, π¦ two is equal to the change in π¦ divided by the change in π₯: π¦ two minus π¦ one over π₯ two minus π₯ one.
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Now, letβs substitute the coordinates in this question into the formula for the slope.
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π¦ two minus π¦ one is equal to 10 minus negative two.
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π₯ two minus π₯ one is equal to negative two minus two.
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This simplifies to 12 over negative four which is equal to negative three.
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So the slope of the line is negative three.
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Substituting this value into our equation for the line gives π¦ is equal to negative three π₯ plus π.
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Next, we need to find the value of π β the π¦-intercept.
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To do this, we can use the fact that either of these two points lie on the straight line and, therefore, their coordinates satisfy the equation of the line.
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Using the first point, this means that when π₯ is equal to two, π¦ is equal to negative two.
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And so we can substitute this pair of values into the equation of the line to give an equation that we can solve for π.
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Substituting two for π₯ and negative two for π¦ gives the equation negative two is equal to negative three multiplied by two plus π.
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Now, we solve for π.
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Negative three multiplied by two is negative six.
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So we have negative two is equal to negative six plus π.
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To solve this equation, we need to add six to both sides.
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This gives four is equal to π.
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Now, we know the value of π, we can substitute this into the equation of our line.
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We have π¦ is equal to negative three π₯ plus four.
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Remember weβre asked for the equation of a straight line in the form ππ₯ plus ππ¦ plus π is equal to zero, so we need all of the terms on the same side.
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We could group them on either side.
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But if I grouped the terms on the left, then the coefficients of both π₯ and π¦ will be positive.
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In order to group the terms on the left-hand side of this equation, I need to add three π₯ to both sides and also subtract four from both sides.
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This gives the equation of the straight line in the requested format: three π₯ plus π¦ minus four is equal to zero.