Math Problem Statement

Garis dengan persamaan ax + by = 8 melalui titik (3, -2) dan (2, 4). Nilai a dan b berturut-turut adalah . . .

Solution

The problem provided is in Indonesian, and it can be translated as:

"A line with the equation ax+by=8ax + by = 8 passes through the points (3,2)(3, -2) and (2,4)(2, 4). The values of aa and bb, respectively, are . . ."

Steps to solve:

We are given the general form of the line equation ax+by=8ax + by = 8 and two points: (3,2)(3, -2) and (2,4)(2, 4). We need to find the values of aa and bb.

  1. Substitute the points into the equation:

    • For the point (3,2)(3, -2): a(3) + b(-2) = 8 \implies 3a - 2b = 8 \tag{1}

    • For the point (2,4)(2, 4): a(2) + b(4) = 8 \implies 2a + 4b = 8 \tag{2}

  2. Solve the system of equations:

    From equation (1): 3a2b=83a - 2b = 8

    From equation (2): 2a+4b=82a + 4b = 8

    Now, we can solve these two simultaneous equations to find aa and bb.

  • Multiply equation (2) by 2 to match the coefficient of aa with that in equation (1): 4a + 8b = 16 \tag{3}

  • Subtract equation (1) from equation (3): (4a+8b)(3a2b)=168(4a + 8b) - (3a - 2b) = 16 - 8 Simplifying: a + 10b = 8 \tag{4}

  1. Solve for aa and bb:

Now, from equation (4):

  • Rearrange: a=810ba = 8 - 10b

  • Substitute this into equation (2): 2(810b)+4b=82(8 - 10b) + 4b = 8 Simplifying: 1620b+4b=8    16b=8    b=1216 - 20b + 4b = 8 \implies -16b = -8 \implies b = \frac{1}{2}

  • Substitute b=12b = \frac{1}{2} into a=810ba = 8 - 10b: a=810(12)=85=3a = 8 - 10\left(\frac{1}{2}\right) = 8 - 5 = 3

Final Answer:

The values of aa and bb are 33 and 12\frac{1}{2}, respectively.

Would you like more details on the solution process or have any other questions?

Here are 5 related questions to consider:

  1. How do you solve a system of linear equations using substitution?
  2. Can you find the equation of the line in slope-intercept form from the two points given?
  3. How does the standard form of a line equation relate to its slope and intercept?
  4. How can you verify that the points (3,2)(3, -2) and (2,4)(2, 4) satisfy the final equation 3x+12y=83x + \frac{1}{2}y = 8?
  5. What are other methods to solve systems of equations, such as elimination?

Tip: Always double-check by substituting the values back into the original equations to confirm the solution is correct.

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Math Problem Analysis

Mathematical Concepts

Linear equations
Simultaneous equations

Formulas

ax + by = c
Substitution method
System of linear equations

Theorems

System of equations can be solved by substitution or elimination methods.

Suitable Grade Level

Grades 9-10