Math Problem Statement

Graph each equation. Determine the solution of the system of equations. x plus 2 yequals4 3 x plus 2 yequals0 Question content area bottom left Part 1 Use the graphing tool to graph the system.

Part 2 What is the solution of the system of​ equations?    enter your response here ​(Type an ordered​ pair.)

Solution

To solve this problem, we will address both parts:

Part 1: Graphing the System of Equations

We are given the system:

  1. x+2y=4x + 2y = 4
  2. 3x+2y=03x + 2y = 0

To graph these, we first rewrite each equation in slope-intercept form (y=mx+by = mx + b):

  1. For x+2y=4x + 2y = 4: 2y=x+4y=12x+22y = -x + 4 \quad \Rightarrow \quad y = -\frac{1}{2}x + 2 This equation has:

    • Slope (mm) = 12-\frac{1}{2}
    • Y-intercept = 22
  2. For 3x+2y=03x + 2y = 0: 2y=3xy=32x2y = -3x \quad \Rightarrow \quad y = -\frac{3}{2}x This equation has:

    • Slope (mm) = 32-\frac{3}{2}
    • Y-intercept = 00 (passes through the origin)

Using these equations, plot both lines on the same graph.


Part 2: Solve for the Intersection Point

The solution to the system is the point where the two lines intersect. To find this algebraically, solve the system:

  1. Start with the equations: x+2y=4(1)x + 2y = 4 \quad \text{(1)} 3x+2y=0(2)3x + 2y = 0 \quad \text{(2)}

  2. Subtract equation (1) from equation (2): (3x+2y)(x+2y)=04(3x + 2y) - (x + 2y) = 0 - 4 2x=4x=22x = -4 \quad \Rightarrow \quad x = -2

  3. Substitute x=2x = -2 into equation (1): 2+2y=42y=6y=3-2 + 2y = 4 \quad \Rightarrow \quad 2y = 6 \quad \Rightarrow \quad y = 3

Thus, the solution is: (2,3)\boxed{(-2, 3)}


Graphing Tool

If you are using a graphing tool, plot:

  • The line y=12x+2y = -\frac{1}{2}x + 2
  • The line y=32xy = -\frac{3}{2}x

Verify that they intersect at (2,3)(-2, 3).


Do you want details or have any questions?

Here are 5 related questions to expand understanding:

  1. How do you graph lines given in standard form without rewriting to slope-intercept form?
  2. How can you solve a system of equations using substitution instead of elimination?
  3. What are the conditions for a system of linear equations to have no solution?
  4. How do slopes of two lines determine the type of solution for a system (parallel, intersecting, or coincident)?
  5. What are some real-world applications of solving systems of linear equations?

Tip: Always double-check calculations when solving systems algebraically to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Graphing Linear Equations

Formulas

Slope-intercept form: y = mx + b
Standard form to slope-intercept form conversion: Ax + By = C → y = -(A/B)x + C/B

Theorems

Intersection of lines in a 2D plane

Suitable Grade Level

Grades 8-10