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Zorluk: ZorSystems of Linear Equations
In the system of linear equations below, aa, bb, and cc are constants.
32x−23y=12ax+by=c\begin{aligned} \frac{3}{2}x - \frac{2}{3}y &= 12 \\ ax + by &= c \end{aligned}
If the system has infinitely many solutions, and the point (4,b)(4, b) lies on the line represented by the second equation, what is the value of cc?
  1. A
    -162
  2. B
    72
  3. C
    108
  4. 162Cevap

Cevap

The correct answer is 162.
The correct answer is 162. Since the system of linear equations has infinitely many solutions, the two equations represent the same line. Therefore, any point that lies on the second line must also lie on the first line. Substituting the coordinates of the point (4, b) into the first equation gives 3/2(4) - 2/3(b) = 12, which simplifies to 6 - 2/3(b) = 12. Solving this equation for b yields b = -9. Since the equations represent the same line, the ratio of their corresponding coefficients must be equal: c / 12 = b / (-2/3). Solving this proportion for c gives c = -18b. Substituting the value of b = -9 into this relation yields c = -18(-9) = 162.

Adım Adım Çözüm

1
Determine the relationship between the two equations based on the number of solutions.
Since the system has infinitely many solutions, the two equations represent coincident lines in the coordinate plane. Therefore, any point that lies on the second line must also satisfy the first equation.
Infinitely many solutions in a system of two linear equations indicate that they represent the same line.
2
Substitute the point (4,b)(4, b) into the first equation to solve for bb.
32(4)−23b=12  ⟹  6−23b=12  ⟹  −23b=6  ⟹  b=−9\frac{3}{2}(4) - \frac{2}{3}b = 12 \implies 6 - \frac{2}{3}b = 12 \implies -\frac{2}{3}b = 6 \implies b = -9
Plugging the coordinates of the point into the first equation allows us to find the value of the unknown coordinate.
3
Set up the ratio of corresponding coefficients for the coincident lines.
a32=b−23=c12\frac{a}{\frac{3}{2}} = \frac{b}{-\frac{2}{3}} = \frac{c}{12}
Equivalent equations must have proportional coefficients and constant terms.
4
Solve for cc using the ratio containing bb and cc.
c12=b−23  ⟹  c=−18b  ⟹  c=−18(−9)=162\frac{c}{12} = \frac{b}{-\frac{2}{3}} \implies c = -18b \implies c = -18(-9) = 162
Substituting the solved value of b=−9b = -9 into the proportion yields the value of cc.

Anahtar Kavram

Systems of Linear Equations with Infinitely Many Solutions
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Konular

Systems of Linear Equations

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