Solve Linear Equations - International Brotherhood of Electrical Workers Exam IBEW

1. Which value of $x$ makes the equation $0x=0$ true?

   All real numbers. The equation simplifies to $0=0$, which holds true for every real number $x$.

2. Identify the solution for $2x+\frac{1}{2}=\frac{7}{2}$.

   $x=\frac{3}{2}$. Subtract $\frac{1}{2}$ from both sides, then divide by 2 to isolate $x$.

3. Identify the solution for $\frac{x}{4}=-3$.

   $x=-12$. Multiply both sides by 4 to eliminate the denominator and isolate $x$.

4. Identify the solution for $7x-5=16$.

   $x=3$. Add 5 to both sides, then divide by 7 to solve for $x$.

5. Identify the solution for $\frac{x}{3}+2=5$.

   $x=9$. Subtract 2 from both sides, then multiply by 3 to solve for $x$.

6. Identify the solution for $\frac{2x-1}{3}=5$.

   $x=8$. Multiply both sides by 3, add 1, then divide by 2 to isolate $x$.

7. Identify the solution for $2(x+5)-3=9$.

   $x=1$. Distribute the 2, subtract 7 from both sides, then divide by 2 to isolate $x$.

8. Identify the solution for $6x-2=2x+14$.

   $x=4$. Subtract $2x$ from both sides, add 2, then divide by 4 to solve for $x$.

9. Identify the solution for $5x=35$.

   $x=7$. Divide both sides by 5 using the multiplication property of equality to solve for $x$.

10. What is the multiplication property of equality used in solving linear equations?

    If $a=b$, then $ac=bc$. This property permits multiplying both sides of an equation by the same non-zero constant to maintain equality, useful for solving equations.

11. Identify the solution for $3x+4=x+10$.

    $x=3$. Subtract $x$ from both sides, subtract 4, then divide by 2 to isolate the variable.

12. Identify the solution for $4(x+1)=20$.

    $x=4$. Divide both sides by 4, then subtract 1 to solve for $x$.

13. Which value of $x$ makes the equation $0x=5$ true?

    No solution. The equation simplifies to $0=5$, a false statement with no value of $x$ satisfying it.

14. Identify the solution for $x+7=12$.

    $x=5$. Subtract 7 from both sides using the addition property of equality to isolate the variable $x$.

15. Identify the solution for $3(x-2)=12$.

    $x=6$. Distribute the 3, add 6 to both sides, then divide by 3 to isolate $x$.

16. What is the distributive property for factoring or expanding expressions?

    $a(b+c)=ab+ac$. This property enables multiplying a term outside parentheses by each term inside, facilitating expansion or factoring in equations.

17. What is the first step to clear fractions in an equation like $\frac{x}{3}+\frac{1}{2}=5$?

    Multiply every term by the LCD. Using the least common denominator turns fractional coefficients into integers, simplifying the equation for easier solving.

18. Identify the solution for $5-2(x-1)=1$.

    $x=3$. Distribute the -2, subtract 7 from both sides, then divide by -2 to solve for $x$.

19. Identify the solution for $2x+3=11$.

    $x=4$. Subtract 3 from both sides, then divide by 2 to isolate $x$.

20. What is the LCD of denominators $2$, $3$, and $6$ for clearing fractions?

    $6$. The least common denominator is the smallest multiple divisible by 2, 3, and 6, enabling fraction clearance in equations.

21. What is the addition property of equality used in solving linear equations?

    If $a=b$, then $a+c=b+c$. This property allows adding the same constant to both sides of an equation without changing the equality, aiding in isolating variables.

22. Identify the solution for $\frac{x}{2}-\frac{x}{3}=1$.

    $x=6$. Combine fractions using LCD 6, then multiply both sides by 6 to solve for $x$.

23. Identify the solution type for $2(x-4)=2x+1$.

    No solution. Distributing and simplifying results in $-8=1$, a contradiction with no satisfying $x$.

24. Identify the solution for $x-9=-2$.

    $x=7$. Add 9 to both sides to isolate $x$, applying the addition property of equality.

25. Identify the solution type for $3(x+1)=3x+3$.

    Infinitely many solutions. Distributing and simplifying yields $3=3$, a true identity for all real $x$.

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