Math Problem Statement

Solve the system of equations by graphing: 1. 2x + y + 10 = 0 2. 6x + y + 26 = 0

Solution

To solve the system of equations given in the image by graphing, we need to rewrite the equations in slope-intercept form (i.e., y=mx+by = mx + b) for easier graphing. Here are the two equations from the image:

  1. 2x+y+10=02x + y + 10 = 0
  2. 6x+y+26=06x + y + 26 = 0

Step 1: Solve for yy in each equation.

For the first equation:

2x+y+10=02x + y + 10 = 0

Subtract 2x2x and 1010 from both sides:

y=2x10y = -2x - 10

For the second equation:

6x+y+26=06x + y + 26 = 0

Subtract 6x6x and 2626 from both sides:

y=6x26y = -6x - 26

Step 2: Graph the equations.

Now that both equations are in the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept:

  • The first equation y=2x10y = -2x - 10 has a slope of 2-2 and a y-intercept of 10-10.
  • The second equation y=6x26y = -6x - 26 has a slope of 6-6 and a y-intercept of 26-26.

Step 3: Find the point of intersection.

You can graph these two lines and find where they intersect. To solve this algebraically:

Set the two expressions for yy equal to each other:

2x10=6x26-2x - 10 = -6x - 26

Solve for xx:

2x+6x=26+10-2x + 6x = -26 + 10 4x=164x = -16 x=4x = -4

Substitute x=4x = -4 into one of the original equations to find yy. Using y=2x10y = -2x - 10:

y=2(4)10=810=2y = -2(-4) - 10 = 8 - 10 = -2

Final Answer:

The solution is (x,y)=(4,2)(x, y) = (-4, -2).

Would you like more details or explanations on any of these steps?


Here are some related questions to deepen your understanding:

  1. How do you interpret the slope of a line graphically?
  2. What are the benefits of solving systems of equations graphically compared to other methods?
  3. How would you verify the solution of a system of equations using substitution?
  4. Can all systems of linear equations be solved by graphing? Why or why not?
  5. How would you interpret a system of equations that has no solution or infinitely many solutions?

Tip: In a system of equations, graphing gives you a visual understanding of how the two lines relate, such as whether they are parallel, intersecting, or coincident (overlapping).

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations
Graphing

Formulas

Slope-intercept form y = mx + b
Substitution method

Theorems

Systems of linear equations and intersections

Suitable Grade Level

Grades 9-11