Math Problem Statement

For the two linear functions, find the point of intersection:

x=y+2 2x−3y=−1

Solution

To find the point of intersection of the two linear functions, we need to solve the system of equations formed by them:

x=y+2x = y + 2 2x3y=12x - 3y = -1

Step 1: Substitute the expression for xx from the first equation into the second equation.

Since x=y+2x = y + 2, substitute this into the second equation:

2(y+2)3y=12(y + 2) - 3y = -1

Step 2: Simplify and solve for yy.

Distribute the 2:

2y+43y=12y + 4 - 3y = -1

Combine like terms:

y+4=1-y + 4 = -1

Subtract 4 from both sides:

y=5-y = -5

Multiply both sides by -1:

y=5y = 5

Step 3: Substitute y=5y = 5 back into the first equation to find xx.

Using the first equation x=y+2x = y + 2:

x=5+2=7x = 5 + 2 = 7

Step 4: Write the solution.

The point of intersection is (x,y)=(7,5)(x, y) = (7, 5).

Final Answer

The two lines intersect at the point (7,5)(7, 5).

Would you like more details or have any questions?

Here are some related questions:

  1. What is the graphical interpretation of the point of intersection?
  2. How would you verify the solution by substituting the values back into the original equations?
  3. How can you solve the system of equations using the elimination method?
  4. What happens if the two lines are parallel?
  5. Can we find the intersection of two linear functions if one or both are vertical lines?

Tip: When solving systems of linear equations, substituting one equation into another is a common method called the substitution method. This technique is particularly useful when one equation is already solved for one variable.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Substitution Method

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12