Math Problem Statement

How many solutions does the system of equations y = 3x - 2 and -3x + y = 7 have: 0, 1, or ∞?

Solution

The system of equations presented in the image is:

y=3x2y = 3x - 2 3x+y=7-3x + y = 7

We are asked how many solutions this system has.

Step 1: Rewrite both equations in slope-intercept form

The first equation is already in slope-intercept form:

y=3x2y = 3x - 2

The second equation, 3x+y=7-3x + y = 7, can be rewritten as:

y=3x+7y = 3x + 7

Step 2: Compare the equations

We now have:

  1. y=3x2y = 3x - 2
  2. y=3x+7y = 3x + 7

Both lines have the same slope (33) but different y-intercepts (2-2 and 77). Since they are parallel and have different y-intercepts, they never intersect.

Conclusion:

The system has 0 solutions because the lines are parallel and do not intersect.

Do you need more details or explanations?

Related Questions:

  1. What is the graphical meaning of parallel lines in a system of equations?
  2. How do we classify systems of equations by their number of solutions?
  3. What is the general approach to solving a system of linear equations?
  4. How can we use substitution to solve a system of equations?
  5. What are consistent, inconsistent, and dependent systems?

Tip:

When solving systems of linear equations, always check if the lines are parallel by comparing their slopes. Parallel lines mean no solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Systems of Equations
Parallel Lines

Formulas

y = mx + b

Theorems

Parallel Line Theorem

Suitable Grade Level

Grades 8-10