Math Problem Statement

What is the solution to y=1/2x-3 and 2x-3y=6

Solution

We are given the system of two linear equations:

  1. y=12x3y = \frac{1}{2}x - 3
  2. 2x3y=62x - 3y = 6

To solve for the values of xx and yy, we can use substitution. Since the first equation already expresses yy in terms of xx, we can substitute this expression for yy into the second equation.

Step 1: Substitute y=12x3y = \frac{1}{2}x - 3 into 2x3y=62x - 3y = 6.

2x3(12x3)=62x - 3\left(\frac{1}{2}x - 3\right) = 6

Step 2: Simplify the equation.

Distribute the 3-3 on the left side:

2x3(12x3)=2x32x+9=62x - 3\left(\frac{1}{2}x - 3\right) = 2x - \frac{3}{2}x + 9 = 6

Step 3: Combine like terms.

2x32x=42x32x=12x2x - \frac{3}{2}x = \frac{4}{2}x - \frac{3}{2}x = \frac{1}{2}x

Now, the equation becomes:

12x+9=6\frac{1}{2}x + 9 = 6

Step 4: Solve for xx.

Subtract 9 from both sides:

12x=69\frac{1}{2}x = 6 - 9 12x=3\frac{1}{2}x = -3

Now multiply both sides by 2 to eliminate the fraction:

x=6x = -6

Step 5: Substitute x=6x = -6 into the first equation to find yy.

Using the equation y=12x3y = \frac{1}{2}x - 3:

y=12(6)3y = \frac{1}{2}(-6) - 3 y=33y = -3 - 3 y=6y = -6

Final Solution:

The solution to the system of equations is:

x=6andy=6x = -6 \quad \text{and} \quad y = -6

Would you like a more detailed explanation or help with other questions?

Related Questions:

  1. How can you solve a system of equations using the elimination method?
  2. What is the graphical interpretation of the solution to a system of linear equations?
  3. How do you check the solution to a system of equations?
  4. What is the slope-intercept form of a linear equation, and how is it used?
  5. Can a system of equations have no solution or infinitely many solutions?

Tip:

When solving systems of equations, always check your solution by substituting the values back into the original equations to confirm they satisfy both equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Substitution Method

Formulas

y = (1/2)x - 3
2x - 3y = 6

Theorems

Substitution method for solving systems of linear equations

Suitable Grade Level

Grades 8-10