Why do equations have the same solution?

Because the lines intersect at a point, there is one solution to the system of equations the lines represent.

Does degree equal number of solutions?

A polynomial equation of degree n has at most n solutions. A polynomial equation of degree n has exactly n solutions, if you count them with multiplicity. The equation x2+1=0 has infinitely many solutions. Indeed, if a,b,c are real numbers satisfying a2+b2+c2=1, then ai+bj+ck is a solution of x2+1=0 in the quaternions.

What does the degree of a polynomial tell you about the number of its roots?

Degree of the Polynomial Remember that the degree of a polynomial, the highest exponent, dictates the maximum number of roots it can have. Thus, the degree of a polynomial with a given number of roots is equal to or greater than the number of roots that are given.

What is the relationship between the degree of a polynomial and the number of real solutions?

The number of solutions of a polynomial equation is always equal to the degree, provided that: We consider complex roots, not just real ones. We count the solutions correctly, including multiplicity.

What are the equation that have the same solution?

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.

How many solutions does each equation have?

If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one solution. If solving an equation yields a statement that is always true, like 3 = 3, then the equation has infinitely many solutions.

What makes an equation have infinite solutions?

If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.

When you can factor expressions using difference of two squares?

When an expression can be viewed as the difference of two perfect squares, i.e. a²-b², then we can factor it as (a+b)(a-b). For example, x²-25 can be factored as (x+5)(x-5). This method is based on the pattern (a+b)(a-b)=a²-b², which can be verified by expanding the parentheses in (a+b)(a-b).

How do you know how many real solutions a polynomial has?

Recall from the Quadratic Functions chapter, that every quadratic equation has two solutions. The degree of a quadratic equation is 2, thus leading us towards the notion that it has 2 solutions. The degree will always tell us the maximum number of solutions a polynomial has.

What is the relationship between the degree of the polynomial?

Give the degree of the polynomial. The degree of an individual term of a polynomial is the exponent of its variable; the exponents of the terms of this polynomial are, in order, 5, 4, 2, and 7. The degree of the polynomial is the highest degree of any of the terms; in this case, it is 7.

How many solutions are in the degree 2 polynomial?

Two
In the following three examples, one can see how these polynomial degrees are determined based on the terms in an equation: y = x (Degree: 1; Only one solution) y = x2 (Degree: 2; Two possible solutions) y = x3 (Degree: 3; Three possible solutions)

Which equation is an example of a higher order equation?

Example: y = 2x + 7 has a degree of 1, so it is a linear equation Example: 5w 2 − 3 has a degree of 2, so it is quadratic Higher order equations are usually harder to solve:

How do you know if two equations have exactly one solution?

If the two equations are in standard form (both variables on one side and a constant on the other side), then the following are true: 1) lf the ratio of the coefficients on the x’s is unequal to the ratio of the coefficients on the y’s (in the same order), then there is exactly one solution.

How do you find equivalent equations in math?

Key Takeaways 1 Equivalent equations are algebraic equations that have identical solutions or roots. 2 Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation. 3 Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.

Why does the equation have an infinite number of solutions?

You now have the same thing on both sides. At this point you can tell that no matter what value you put in for x, the equation will always be true. That is why it has an infinite number of solutions. What Sal did is went ahead and solved it by eliminating the x terms. So what Sal is saying is that 2=2 no matter what x you input.