Solve Linear Systems - International Brotherhood of Electrical Workers Exam IBEW

1. Solve by elimination: $x+2y=8$ and $3x-2y=4$.

   $(3,\frac{5}{2})$. Adding eliminates y, giving $4x=12$ so $x=3$, then $2y=5$ so $y=\frac{5}{2}$.

2. Which point is the solution to $x+y=5$ and $x-y=3$?

   $(4,1)$. Adding gives $2x=8$ so $x=4$, subtracting gives $2y=2$ so $y=1$.

3. What is the intersection point of the lines $x=2$ and $y=-3$?

   $(2,-3)$. The vertical line $x=2$ and horizontal line $y=-3$ cross at this unique point.

4. Solve the system: $x-2y=0$ and $x+y=9$.

   $(6,3)$. From $x=2y$, substitute into second to get $3y=9$ so $y=3$, then $x=6$.

5. Identify the method best suited if one equation is already solved for a variable, such as $y=3x-2$.

   Substitution. With one variable isolated, direct replacement into the other equation simplifies solving.

6. What result indicates a system is inconsistent after elimination (no solution)?

   A false statement such as $0=5$. A contradiction after elimination shows the system is inconsistent with no satisfying values.

7. Find the missing multiplier $k$ so that multiplying $3x-2y=7$ by $k$ makes the $x$-coefficients cancel with $6x+y=1$.

   $k=-2$. To oppose the x-coefficient $6$, set $3k=-6$, solving to $k=-2$ for cancellation.

8. Identify the solution type: $y=2x+3$ and $y=2x-1$.

   No solution. Same slope $2$ but different y-intercepts indicate parallel lines with no intersection.

9. What is a quick way to check a proposed solution $(x,y)$ for a system?

   Substitute into both equations; both must be true. Plugging in verifies the point satisfies both original equations, confirming it's the solution.

10. What is the standard form of a linear equation often used for elimination?

    $Ax+By=C$. This form aligns coefficients of x and y for easy addition or subtraction in elimination.

11. What does it mean if a system of two linear equations has no solution?

    The lines are parallel (same slope, different intercept). Parallel lines with the same slope but different y-intercepts never intersect, resulting in no common solution.

12. Solve by substitution: $x=5-y$ and $2x+y=11$.

    $(6,-1)$. Substituting $x=5-y$ into $2x+y=11$ gives $10-y=11$ so $y=-1$, then $x=6$.

13. What is the elimination method for solving a system of two linear equations?

    Add/subtract equations to eliminate one variable. Combining equations cancels one variable, allowing solution for the remaining one then back-substitution.

14. What is the slope-intercept form of a line used to compare slopes in a system?

    $y=mx+b$. This form directly shows slope $m$ for comparison and y-intercept $b$ in systems.

15. Identify the method best suited if equations are in $Ax+By=C$ form with easy opposites, such as $x+2y=5$ and $-x+y=1$.

    Elimination. Opposite coefficients for a variable allow immediate cancellation when adding the equations.

16. What is the substitution method for solving a system of two linear equations?

    Solve one equation for a variable, substitute into the other. Isolating one variable and replacing it in the other equation reduces the system to one variable.

17. Identify the solution type: $2x+4y=6$ and $x+2y=3$.

    Infinitely many solutions. The second equation is half the first, so they represent the same line with all points as solutions.

18. What does it mean if a system of two linear equations has infinitely many solutions?

    The equations represent the same line (equivalent equations). When equations are multiples of each other, they describe the same line with all points as solutions.

19. What is the ordered-pair form used to report a solution to a two-variable system?

    An ordered pair $(x,y)$. This format specifies the unique x and y values satisfying both equations simultaneously.

20. Solve the system by inspection: $x+y=7$ and $x-y=1$.

    $(4,3)$. Adding the equations eliminates y, giving $2x=8$ so $x=4$, then $y=3$.

21. What result indicates dependent equations after elimination (infinitely many solutions)?

    A true statement such as $0=0$. An identity indicates the equations are dependent, sharing infinitely many solutions along the line.

22. Solve by elimination: $2x+y=9$ and $2x-y=3$.

    $(3,3)$. Adding eliminates y, yielding $4x=12$ so $x=3$, then substitution gives $y=3$.

23. Solve by substitution: $y=2x+1$ and $x+y=10$.

    $(3,7)$. Substituting $y=2x+1$ into $x+y=10$ gives $3x=9$ so $x=3$, then $y=7$.

24. What is the first step of elimination if no variable coefficients are opposites?

    Multiply one or both equations to create opposite coefficients. Scaling creates additive inverses for a variable's coefficients, enabling elimination upon addition.

25. What does it mean if a system of two linear equations has exactly one solution?

    The lines intersect at exactly one point $(x,y)$. This occurs when the lines are not parallel and not coincident, intersecting at a unique point.

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