Weighted Equations - Examples, Exercises and Solutions

Equivalent Equations

Equivalent equations are different equations whose basically equal to each other, as they share the same variable, in addition to sharing the same solution. That means that if two equations simplify to the same solution when solved, they are equivalent.

We can easily move from one equivalent equation to another. In fact, each equation has an infinity of equivalent equations. This is due to the following: if you add, subtract, multiply, or divide both sides of an equation by a number different from zero, another equation equivalent to the initial one is obtained.

The importance of equivalent equations

By using equivalent equations, we can basically change the equation step by step, without changing it solution. In other words, using equivalent equations to maintain equality, helps us simplify complex expressions or isolate variables.

X=-6 Equivalent Equations

Suggested Topics to Practice in Advance

  1. What is the unknown of a mathematical equation?
  2. Numerical Value
  3. Variable
  4. Equations
  5. Solution of an equation

Practice Weighted Equations

Examples with solutions for Weighted Equations

Exercise #1

Are the equations balanced?

9x=1=?5x=40 9-x=1\stackrel{?}{=}5x=-40

Video Solution

Step-by-Step Solution

To determine if the two equations are balanced, we need to evaluate them separately:

First, we solve the equation 9x=19 - x = 1.

  • Subtract 9 from both sides to isolate x-x:
  • x=19-x = 1 - 9
  • x=8-x = -8
  • Multiply both sides by 1-1 to solve for xx:
  • x=8x = 8

Next, we solve the equation 5x=405x = -40.

  • Divide both sides by 5 to isolate xx:
  • x=405x = \frac{-40}{5}
  • x=8x = -8

The solution to the first equation is x=8x = 8 and the solution to the second equation is x=8x = -8. Since the solutions for xx are not the same, the two equations are not balanced.

Therefore, the correct answer is No.

Answer

No

Exercise #2

Are the equations balanced?

26=x6=?2x=178 26{=}\frac{x}{6}\stackrel{?}{=}\frac{2}{x}{=}\frac{1}{78}

Video Solution

Step-by-Step Solution

To determine whether the given equations are balanced, we begin by examining them step-by-step:

  • Given: 26=x6=?2x=17826 = \frac{x}{6} \stackrel{?}{=} \frac{2}{x} = \frac{1}{78}

We should first determine the value of xx from the equation involving 178\frac{1}{78}:

  • The last equation is 178=2x\frac{1}{78} = \frac{2}{x}.
  • Multiplying both sides by xx and then by 7878 gives x=156x = 156.

Use x=156x = 156 to evaluate the other expressions:

  • For x6\frac{x}{6}:
  • 1566=26\frac{156}{6} = 26

Now, check if all values equate:

  • 26=2626 = 26 from x6\frac{x}{6}
  • 26=2626 = 26 from the right side of the original problem statement

All values match, indicating:

The equations are balanced. Therefore, the correct answer is Yes.

Answer

Yes

Exercise #3

Are the equations balanced?

x25=5=?7x10=200 x-25=5\stackrel{?}{=}7x-10=200

Video Solution

Step-by-Step Solution

We will solve each equation step-by-step to ascertain if there is a common solution indicative of them being balanced:

Solving the first equation x25=5 x - 25 = 5 :

  • Add 25 to both sides to isolate x x :
x25+25=5+25 x - 25 + 25 = 5 + 25 x=30 x = 30

Solving the second equation 7x10=200 7x - 10 = 200 :

  • Add 10 to both sides to isolate the term with x x :
7x10+10=200+10 7x - 10 + 10 = 200 + 10 7x=210 7x = 210
  • Divide by 7 to solve for x x :
x=2107 x = \frac{210}{7} x=30 x = 30

Both equations give the solution x=30 x = 30 . This indicates that they are indeed balanced, as they share a common solution for x x .

Therefore, the equations are balanced.

The correct answer to the problem is: Yes.

Answer

Yes