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If the slope of the straight line three ๐ plus seven all multiplied by ๐ฅ plus four ๐๐ฆ plus four is equal to zero equals negative one, find the value of ๐.
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In this question, weโre given the equation of a straight line, and weโre told that the slope of this straight line is equal to negative one.
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We need to use this information to determine the value of ๐.
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To answer this question, we need to notice some things: first, the form weโre given the equation of our straight line in.
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This is called the general form of the equation of a straight line, and itโs not easy to find the slope from this form.
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However, we do know a form of straight line where itโs easy to find the slope of our line.
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We know the slopeโintercept form of a straight line is the equation ๐ฆ is equal to ๐๐ฅ plus ๐, where ๐ is the slope of our straight line and ๐ is the ๐ฆ-intercept of our straight line.
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If we could rearrange the equation for the straight line given to us into this form, then we know that the coefficient of ๐ฅ must be the slope.
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It must be equal to negative one.
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And in fact, we can almost always do this.
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The only time this will be complicated is if weโre dealing with a vertical line because when weโre dealing with a vertical line, we wonโt have a value for the slope.
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However, in this case, we know the slope is equal to negative one, so we donโt need to worry about these two cases.
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This means we know we can always find the equation in this form.
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To do this, we want ๐ฆ to be on its own on the left-hand side of our equation.
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So letโs start by subtracting both terms which donโt involve ๐ฆ.
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So we subtract three ๐ plus seven times ๐ฅ from both sides of our equation, and we subtract four from both sides of our equation.
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This gives us the equation four ๐๐ฆ is equal to negative one times three ๐ plus seven multiplied by ๐ฅ minus four.
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Next, in our slopeโintercept form, the coefficient of ๐ฆ needs to be one.
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So weโre going to need to divide through both sides of our equation through by four ๐.
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And itโs worth reiterating here we know the value of ๐ is not equal to zero because if our value of ๐ was equal to zero, we would lose the ๐ฆ-term in our general equation for the straight line.
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Then, if we were to solve this equation, we will get one solution for ๐ฅ and we could have any value for ๐ฆ.
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In other words, we get a vertical line.
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But we know that we donโt have a vertical line because we know the slope is equal to negative one.
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So we divide both sides of our equations through by four ๐.
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We get ๐ฆ is equal to negative one times three ๐ plus seven multiplied by ๐ฅ minus four all divided by four ๐.
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But we want to write this in slopeโintercept form.
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So weโre going to split this in our numerator.
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This gives us that ๐ฆ is equal to negative one times three ๐ plus seven all over four ๐ all multiplied by ๐ฅ minus four divided by four ๐.
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And we can simplify this although itโs not necessary; in our second term, four divided by four is equal to one.
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Remember, in the slopeโintercept form of a straight line, the coefficient of ๐ฅ is the slope of our line, and weโre told in the question this is equal to negative one.
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Therefore, we can get an equation for ๐ if we just set the coefficient of ๐ฅ equal to negative one; we get the equation negative one is equal to negative one times three ๐ plus seven all divided by four ๐.
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Now all we need to do is solve this equation for ๐.
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Weโll start by multiplying through by four ๐.
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We get negative four ๐ is equal to negative one times three ๐ plus seven.
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Next, weโre going to distribute the negative over our parentheses.
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This gives us negative four ๐ is equal to negative three ๐ minus seven.
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Next, weโre going to add three ๐ to both sides of our equation.
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This gives us negative four ๐ plus three ๐ is equal to negative seven.
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We know that negative four plus three is equal to negative one.
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So this equation simplifies to give us that negative ๐ is equal to negative seven and we can solve for our value of ๐ by multiplying through by negative one.
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We get that ๐ is equal to seven, which is our final answer.
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Therefore, given that the slope of the straight line three ๐ plus seven all multiplied by ๐ฅ plus four ๐๐ฆ plus four is equal to zero was equal to negative one, then, by writing this equation in slopeโintercept form, we were able to show that the value of ๐ had to be equal to seven.