Math Problem Statement

-x+2y=6 and y=1/2x+3 Graph the system below and write its solution. Note that you can also answer "No solution" or "Infinitely many" solutions.

Solution

Let's first graph the system of equations and determine their solution.

Given System of Equations:

  1. x+2y=6-x + 2y = 6
  2. y=12x+3y = \frac{1}{2}x + 3

Step 1: Rewrite the first equation in slope-intercept form (y = mx + b)

We want to express the first equation in a form similar to the second one so it's easier to graph.

Start with the equation: x+2y=6-x + 2y = 6 First, solve for yy. Add xx to both sides: 2y=x+62y = x + 6 Now divide everything by 2: y=12x+3y = \frac{1}{2}x + 3

Step 2: Analyze the equations

Both equations now have the same form:

  1. y=12x+3y = \frac{1}{2}x + 3
  2. y=12x+3y = \frac{1}{2}x + 3

Step 3: Graph the equations

Both equations represent straight lines with the same slope (12\frac{1}{2}) and the same y-intercept (3). Therefore, the two lines are identical.

Conclusion:

Since the two lines are the same, the system has infinitely many solutions (every point on the line is a solution).

Would you like to see a graph of the system, or would you like to explore any further details?


Here are some follow-up questions for you:

  1. What would happen if the slopes were different in the two equations?
  2. How do you solve systems with two lines that have no solution?
  3. How can you tell when two lines are parallel just from the equations?
  4. How do you solve a system of equations graphically?
  5. What other methods can be used to solve systems of linear equations, besides graphing?

Tip: When two lines have the same slope and y-intercept, they are coincident lines, meaning they overlap entirely, and the system has infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Graphing Systems of Equations

Formulas

y = mx + b
Linear equation manipulation

Theorems

Coincident lines theorem (infinitely many solutions when lines are identical)

Suitable Grade Level

Grades 8-10