Algebra 1 problems

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Analyzing Systems of Linear Equations

In the given exercises, the task is to match the system of linear equations with its graph and determine whether it has one solution, no solution, or infinitely many solutions. Let’s explore each system.

Exercise 3

System of Equations:

  1. -x + y = 1

  2. x - y = 1

  3. Solving the System:

    Equation 1: -x + y = 1

    Solving for y gives us: y = x + 1.

    Equation 2: x - y = 1

    Solving for y gives us: y = x - 1.

  4. Graphical Representation:

    • These are two lines with the same slope, but different y-intercepts (1 and -1).
    • The lines are parallel and do not intersect.

  5. Conclusion: No Solution

    This system has no solution as the lines are parallel.

Exercise 4

System of Equations:

  1. 2x - 2y = 4

  2. -x + y = -2

  3. Solving the System:

    Equation 1: 2x - 2y = 4

    Dividing the entire equation by 2 gives us:

    x - y = 2

    Equation 2: -x + y = -2

    Simplifying gives: x - y = 2.

  4. Graphical Representation:

    • These equations are equivalent.
    • They represent the same line.

  5. Conclusion: Infinitely Many Solutions

    There are infinitely many solutions because the equations represent the same line.

Exercise 5

System of Equations:

  1. 2x + y = 4

  2. -4x - 2y = -8

  3. Solving the System:

    Equation 1: 2x + y = 4

    Solving for y gives us: y = -2x + 4.

    Equation 2: -4x - 2y = -8

    Rearranging gives: 2y = -4x + 8, or y = -2x + 4.

  4. Graphical Representation:

    • These are not distinct equations; they are multiples.

  5. Conclusion: Infinitely Many Solutions

    The second equation is a multiple of the first, so they are the same line.

Exercise 6

System of Equations:

  1. x - y = 0

  2. 5x - 2y = 6

  3. Solving the System:

    Equation 1: x - y = 0

    Solving for y gives us: y = x.

    Equation 2: 5x - 2y = 6

    Substituting y = x gives: 5x - 2x = 6, or 3x = 6, solving for x, x = 2.

    Substituting back: y = 2.

  4. Graphical Representation:

    • The second equation’s line crosses the first at the point (2, 2).

  5. Conclusion: One Solution

    This system has exactly one solution: (2, 2).

Exercise 7

System of Equations:

  1. -2x + 4y = 1

  2. 3x - 6y = 9

  3. Solving the System:

    Equation 1: -2x + 4y = 1

    Rearrange to find y:

    4y = 2x + 1 \ \text{,(incorrect coefficient)}

    Rearranging, divide by 4 gives y = \frac{1}{2}x +\frac{1}{4}

    Equation 2: 3x - 6y = 9

    Rearrange to find y:

    3x - 6y = 9 \implies 3x = 6y + 9

    Rearrange to y = \frac{1}{2}x - 1.5

  4. Graphical Representation:

    • These lines have the same slope but different y-intercepts.

  5. Conclusion: No Solution

    The lines are parallel and never intersect.

Exercise 8

System of Equations:

  1. 5x + 3y = 17

  2. x - 3y = -2

  3. Solving the System:

    Equation 1: 5x + 3y = 17

    Rearrange to find y:

    3y = -5x + 17

    Equation 2: x - 3y = -2

    Rearrange to find y:

    $3y = x + 2

  4. Graphical Representation:

    • These are two distinct lines.

  5. Conclusion: One Solution

    The lines will intersect at one point only.

By solving these problems, you should have a clear understanding of how to determine the nature of solutions from systems of linear equations! Each system has been systematically evaluated to determine its solution type, offering insight into algebra and graphing. @Blibloop