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If the line passing through points π΄: negative thirteen, eight and π΅: twenty, π¦ is parallel to the line passing through points πΆ: negative two, zero and π·: seven, π¦, what is the value of π¦?
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For two lines to be parallel, they must have the exact same slope.
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They must be equal.
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So in order for the slopes to be the same, we would need to set the slope formulas equal to each other for each line.
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So to find the slope of a line, itβs the change in π¦ divided by the change in π₯, so π¦ two minus π¦ one over π₯ two minus π₯ one.
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Itβs also commonly written as π¦ one minus π¦ two over π₯ one minus π₯ two.
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As long as you follow that same pattern, either formula would be fine.
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Here we have our points.
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π΄ and π΅ create a line as well as πΆ and π·; they create a line.
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Now weβll label our points.
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π΄ the negative thirteen is π₯ one and eight is the π¦ one and π΅ thatβs our second point thatβs listed, so twenty is π₯ two and π¦ is our π¦ two, and same thing with πΆ and π·.
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Letβs first begin by finding the slope of π΄π΅, so π¦ two minus π¦ one, so π¦ minus eight, over π₯ two minus π₯ one, so twenty minus negative thirteen.
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So really thatβs adding.
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So on the top, we have π¦ minus eight, which doesnβt simplify.
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And on the bottom, twenty plus thirteen is equal to thirty three.
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Now letβs do the same thing for line πΆπ·.
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So for slope of πΆπ·, we would take π¦ two minus π¦ one, so π¦ minus zero, over π₯ two minus π₯ one, so seven minus negative two.
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So the slope of πΆπ· will be π¦ minus zero all over nine.
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Now these lines are parallel, which means they should have the exact same slope.
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So we can take both of these fractions and set them equal to each other, and we can take these again because these lines are supposed to be parallel.
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So the slope should be equal.
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So if we set them equal to each other, we can find the cross product and solve for π¦.
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So here weβre setting our slopes equal to each other and now we will cross-multiply.
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So we have nine times π¦ minus eight equal to thirty three times π¦ minus zero.
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Now we distribute.
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So we will take nine times π¦ and then nine times negative eight and thirty three times π¦ and thirty three times zero.
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So we have nine π¦ minus seventy two equals thirty three π¦.
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The minus zero doesnβt really do anything.
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So now we need to isolate the π¦.
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So letβs go ahead and add nine π¦ to both sides of the equation.
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Doing so, the nine π¦s cancel.
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Weβve negative seventy two left on the left-hand side of the equation.
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And on the right-hand side, thirty three π¦ plus nine π¦ is twenty four π¦.
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Now our last step would be to divide both sides by twenty four, which means π¦ is equal to negative three.
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So the value of π¦ knowing that these lines are parallel, we take the slopes, we set them equal to each other, and then we solved for π¦.
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And we got that π¦ is equal to negative three.