Math Problem Statement

y=-2x+8 y=-4x+16

Solution

Let's solve the system of linear equations:

  1. y=2x+8y = -2x + 8
  2. y=4x+16y = -4x + 16

Step 1: Set the equations equal to each other.

Since both equations equal yy, we set them equal: 2x+8=4x+16-2x + 8 = -4x + 16

Step 2: Solve for xx.

Simplify the equation: 2x=82x = 8 x=4x = 4

Step 3: Solve for yy using x=4x = 4.

Substitute x=4x = 4 into either equation (let's use the first one): y=2(4)+8y = -2(4) + 8 y=8+8y = -8 + 8 y=0y = 0

Solution:

The solution to the system is: (x,y)=(4,0)(x, y) = (4, 0)

Verification:

Substitute x=4x = 4 into the second equation to ensure consistency: y=4(4)+16y = -4(4) + 16 y=16+16y = -16 + 16 y=0y = 0 The solution is verified.


Would you like further details or have any questions about this process?
Here are 5 related questions to explore:

  1. How do you graphically interpret the solution to this system of equations?
  2. Can you identify if the system is consistent, inconsistent, or dependent?
  3. What happens if we modify one of the equations slightly (e.g., y=4x+15y = -4x + 15)?
  4. How would the solution differ if the two lines were parallel?
  5. Can we solve this system using matrices or another algebraic method?

Tip: Always verify solutions by substituting them back into the original equations!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
System of Equations

Formulas

Equation of a line: y = mx + b
Setting equations equal to solve for x and y

Theorems

Equality of two expressions for solving systems of equations

Suitable Grade Level

Grades 8-10