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The slope of line n in the xy-plane is not 0 and the y-inter [#permalink]

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12 Jan 2014, 02:56

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The slope of line n in the xy-plane is not 0 and the y-intercept of line n in the xy-plane is not 0. The equation of line n is y = mx + b, where m and b are constants. What is the value of m + b?

(1) The y-intercept is equal to the square of the x-intercept. (2) The line goes through the point (−8, −16).

(1): By definition, y-intercept of a line in form y = mx + b is equal to b. x-intercept is value of x when y = 0. y = mx + b → 0 = mx + b → mx = −b → x = -(b/m) Thus, y-intercept is b & x-intercept is –(b/m). Since y-intercept is square of x-intercept, we can create equation: b = (-b/m)^2 → m^2*b= b^2 Since we are told in question stem that b does not equal 0, we can divide both sides by b. m^2= b Since we know that b is equal to m^2, we can substitute that value into equation of line n. y = mx + b → y = mx + m^2 "what is value of m + b?" Different values of m lead to different values for b. So different values for m lead to different values for m + b. For example, if m = 1, then b = 1^2 = 1 → m + b = 1 + 1 = 2 If m = 2, then b = m^2 = 4 → m + b = 2 + 4 = 6 Insufficient.

(2): In order to determine equation of a line, it's necessary to have either two points, or one point and slope. Therefore, it is impossible to derive an equation of a line from simply one point on line. There are infinitely many different lines that go through point (−8, −16). So there are infinitely many different possible values of m + b. Insufficient.

Combined: From (1), equation of line n is of form y = mx + m^2, where m is slope and m≠ 0. (2) says that line goes through point (−8, −16). Remember that any paired point in form (x, y) represents a pair of values that could satisfy equation. In other words, first term in ordered pair can substitute in for x, and second term can be substituted in for y. Substituting −8 for x and −16 for y, we have: −16 = m(−8) + m^2 → −16 = −8m + m^2 → 16 − 8m + m^2 = 0 With quadratics, it's difficult to tell sufficiency without solving. This is because sometimes quadratics yield a single repeated root, and sometimes they yield two different values. m^2 − 8m + 16 = (m − 4)(m − 4) = (m − 4)^2 If (m − 4)^2 = 0, then m = 4 As b = m^2 → b = 4^2 → b = 16 Able to determine m + b = 4 + 16 = 20 Sufficient.

Hi, I want to know if we have simpler solution for this question, please.

The slope of line n in the xy-plane is not 0 and the y-intercept of line n in the xy-plane is not 0. The equation of line n is y = mx + b, where m and b are constants. What is the value of m + b?

(1) The y-intercept is equal to the square of the x-intercept. (2) The line goes through the point (−8, −16).

(1): By definition, y-intercept of a line in form y = mx + b is equal to b. x-intercept is value of x when y = 0. y = mx + b → 0 = mx + b → mx = −b → x = -(b/m) Thus, y-intercept is b & x-intercept is –(b/m). Since y-intercept is square of x-intercept, we can create equation: b = (-b/m)^2 → m^2*b= b^2 Since we are told in question stem that b does not equal 0, we can divide both sides by b. m^2= b Since we know that b is equal to m^2, we can substitute that value into equation of line n. y = mx + b → y = mx + m^2 "what is value of m + b?" Different values of m lead to different values for b. So different values for m lead to different values for m + b. For example, if m = 1, then b = 1^2 = 1 → m + b = 1 + 1 = 2 If m = 2, then b = m^2 = 4 → m + b = 2 + 4 = 6 Insufficient.

(2): In order to determine equation of a line, it's necessary to have either two points, or one point and slope. Therefore, it is impossible to derive an equation of a line from simply one point on line. There are infinitely many different lines that go through point (−8, −16). So there are infinitely many different possible values of m + b. Insufficient.

Combined: From (1), equation of line n is of form y = mx + m^2, where m is slope and m≠ 0. (2) says that line goes through point (−8, −16). Remember that any paired point in form (x, y) represents a pair of values that could satisfy equation. In other words, first term in ordered pair can substitute in for x, and second term can be substituted in for y. Substituting −8 for x and −16 for y, we have: −16 = m(−8) + m^2 → −16 = −8m + m^2 → 16 − 8m + m^2 = 0 With quadratics, it's difficult to tell sufficiency without solving. This is because sometimes quadratics yield a single repeated root, and sometimes they yield two different values. m^2 − 8m + 16 = (m − 4)(m − 4) = (m − 4)^2 If (m − 4)^2 = 0, then m = 4 As b = m^2 → b = 4^2 → b = 16 Able to determine m + b = 4 + 16 = 20 Sufficient.

Hi, I want to know if we have simpler solution for this question, please.

The slope of line n in the xy-plane is not 0 and the y-intercept of line n in the xy-plane is not 0. The equation of line n is y = mx + b, where m and b are constants. What is the value of m + b?

(1) The y-intercept is equal to the square of the x-intercept. The y-intercept is the value of y for x=0, so b, and the x-intercept is the value of x for y=0, so -b/m. Thus we are given that \(b=\frac{b^2}{m^2}\) --> reduce by b, which we know is not 0 to get: \(m^2=b\). Not sufficient to get the value of m+b.

(2) The line goes through the point (−8, −16). Clearly insufficient: one point of a line cannot give neither the slope nor the y-intercept of it.

(1)+(2) From (2) we have that \(-16=-8m+b\) and from (1) we have that \(m^2=b\) --> substitute: \(-16=-8m+m^2\) --> \((m-4)^2=0\) --> \(m=4\) --> \(b=m^2=16\) --> \(m+b=20\). Sufficient.

Re: The slope of line n in the xy-plane is not 0 and the y-inter [#permalink]

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22 Oct 2017, 01:44

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