Math Problem Statement

Tentukan x dan y

Tentukan x dan y

Solution

The image contains a system of inequalities and matrix expressions involving xx and yy. Based on the text, I see the following:

  1. Inequality: x+y8-x + y \geq 8

  2. Matrix equation: (2x5+2y)=(107)\left( \begin{array}{c} 2 - x \\ 5 + 2y \end{array} \right) = \left( \begin{array}{c} 10 \\ -7 \end{array} \right)

We can solve this system step by step:

Step 1: Solve the matrix equation

From the matrix equation: 2x=10and5+2y=72 - x = 10 \quad \text{and} \quad 5 + 2y = -7

Solve each equation:

  • For 2x=102 - x = 10, subtract 2 from both sides: x=8x=8-x = 8 \quad \Rightarrow \quad x = -8

  • For 5+2y=75 + 2y = -7, subtract 5 from both sides: 2y=12y=62y = -12 \quad \Rightarrow \quad y = -6

Step 2: Check inequality

Substitute x=8x = -8 and y=6y = -6 into the inequality: x+y8(8)+(6)8868-x + y \geq 8 \quad \Rightarrow \quad -(-8) + (-6) \geq 8 \quad \Rightarrow \quad 8 - 6 \geq 8 This simplifies to: 282 \geq 8 This inequality is false, meaning the solution does not satisfy the inequality.

Thus, the system does not have a valid solution that satisfies both the matrix equation and the inequality.

Would you like more details or further assistance?

Related questions:

  1. What is the solution of the system if we only consider the matrix equation?
  2. How can we analyze the inequality's meaning geometrically?
  3. Can we modify the inequality to make the solution valid?
  4. How would the solution change if we solved a similar system with different constants?
  5. How are systems of equations and inequalities applied in optimization problems?

Tip: In systems involving both equations and inequalities, it’s often helpful to solve the equations first and check if the results satisfy the inequality.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
System of Equations
Matrices

Formulas

Matrix equation: \left( \begin{array}{c} 2 - x \\ 5 + 2y \end{array} \right) = \left( \begin{array}{c} 10 \\ -7 \end{array} \right)
Inequality: -x + y \geq 8

Theorems

Solving a system of linear equations using substitution

Suitable Grade Level

Grade 10-12