Do equivalent equations have the same solution?
Raising both sides of the equation to the same odd power or taking the same odd root will produce an equivalent equation. If both sides of an equation are non-negative, raising both sides of an equation to the same even power or taking the same even root will give an equivalent equation.
What are two equations that have the same solution set?
Systems of equations that have the same solution are called equivalent systems. Given a system of two equations, we can produce an equivalent system by replacing one equation by the sum of the two equations, or by replacing an equation by a multiple of itself.
What solution means that the system of equations has the same equation?
Systems of equations can be classified by the number of solutions. If a system has at least one solution, it is said to be consistent . If a consistent system has exactly one solution, it is independent .
What is it called when both sides of an equation are equal?
Sometimes people refer to this as keeping the equation “balanced”. If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced. Let’s look at a simple numeric equation, 3 + 7 =10, to explore the idea of an equation as being balanced.
Is an equation that is satisfied for all values of the variable for which both sides of the equation are defined?
An equation that is satisfied for every value of the variable for which both sides of the equation are defined is called an identity. Conditional equations are true for some values of the variable and false for others.
What kind of system of linear equations is shown if they have equal slopes and equal y intercepts?
A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, the lines coincide so the equations represent the same line.
Which systems of equations have infinite solutions?
An independent system of equations has exactly one solution (x,y) . An inconsistent system has no solution, and a dependent system has an infinite number of solutions.
Why do we do the same thing to both sides of an equation?
It comes down to what the equals sign means. It means, as @MaliceVidrine says, that the two things mentioned on either side of the equation are the same thing. If you do something to the left side, you have presumably changed it.
When you square each side of an equation do you always get an equivalent equation?
14 Answers. If two things are equal, then so long as you do the same thing to both, they will remain equal. There is nothing wrong with taking the square of both sides of an equation.
Is an equation that is false for every value of the variable?
An equation that is false for every value of the variable is a contradiction. An equation that is satisfied for every value of the variable for which both sides of the equation are defined is called an identity.
How do you know if two systems of equations are equivalent?
Given a system of two equations, we can produce an equivalent system by replacing one equation by the sum of the two equations, or by replacing an equation by a multiple of itself. In contrast, we can be sure that two systems of equations are not equivalent if we know that a solution of the one is not a solution of the other.
Which is a solution of one of the two systems?
We can pre-multiply both sides of the last equation by and obtain or Since , we have Thus, we have proved that which, together with the implication derived previously, gives that is, is a solution of one of the two systems if and only if it is a solution of the other one. In other words, the two systems are equivalent.
How do you solve a set of equations with the same solution?
We could solve both equations to see if they have the same solution set. Subtract x from both sides and subtract 5 from both sides. Divide both sides by 3. Add 5 to both sides. Distribute the 3 to remove the parentheses. Subtract 21 x from both sides. Divide both sides by -9. Interchange sides.
What is the difference between the first and the second equation?
The only difference between the first equation in each set is that the first one is three times the second one (equivalent). The second equation is exactly the same. Helmenstine, Anne Marie, Ph.D. “Understanding Equivalent Equations in Algebra.”