# How To Solve Systems of Nonlinear Equations

TLDRThis video script offers a comprehensive guide to solving systems of non-linear equations. It begins by demonstrating the verification of solutions for two linear equations and then transitions to solving more complex systems using elimination and substitution methods. The script covers various examples, including systems with quadratic equations, and illustrates the process of simplifying and factoring to find solutions. It also emphasizes the importance of checking each solution to ensure its validity. The tutorial concludes with a systematic approach to solving equations involving squares and square roots, resulting in multiple possible solutions, all of which are verified for correctness.

###### Takeaways

- π When solving systems of equations, not all possible solutions may be correct, and it's important to check them against the original equations.
- π The first example demonstrates checking solutions by substituting values into both equations to ensure they satisfy the conditions.
- π’ In the second example, elimination is used to solve a system of nonlinear equations, showing how to manipulate equations to isolate variables.
- π The process of elimination involves combining equations to cancel out variables, simplifying the system to a single variable equation.
- β After finding a solution for one variable, it's plugged back into the original equations to find the corresponding value for the other variable.
- π The third example uses substitution to solve a system involving quadratic equations, highlighting the process of solving one equation for one variable and substituting into the other.
- π The concept of factoring is introduced to solve a quadratic equation, finding two numbers that meet specific multiplication and addition criteria.
- π€ The script suggests that for each solution found, it's crucial to verify its correctness by plugging it back into the original equations.
- π The fourth example involves a system of equations that can be solved using both substitution and elimination, emphasizing the flexibility in choosing a method.
- π§© The final example with x^2 + y^2 and x^2 - y^2 equations shows how to use elimination to simplify the system, leading to finding all possible solutions for x and y.
- π The video concludes by summarizing the process of solving systems of nonlinear equations, reinforcing the importance of checking each solution for accuracy.

###### Q & A

### What is the first step in verifying a solution for a system of equations according to the script?

-The first step is to replace the variables with the proposed solution values and check if the left-hand side equals the right-hand side of the equations.

### How does the script suggest checking the correctness of a solution for the equations x + y = 3 and y = x^2 + 1?

-By substituting x with -2 and y with 5 into both equations to see if they satisfy the conditions.

### What method does the script use to solve the system of nonlinear equations y = x^2 + 3x - 1 and y = x^2 - 4x + 6?

-The script uses the elimination method by multiplying the second equation by -1 and adding the two equations to eliminate the y variable.

### How does the script simplify the equation 3x - 7 = 0 after using elimination on the nonlinear system?

-By moving -7 to the other side to get 3x = 7 and then dividing by 3 to find x = 1.

### What substitution does the script make to solve for y in the equation x - y = -2?

-The script substitutes y with x + 2 after moving the -2 to the other side and changing it to +2.

### How does the script expand the binomial (x + 2)^2 when solving the system x^2 + y^2 = 20 and x - y = -2?

-The script expands it to x^2 + 4x + 4 by using the distributive property (x * x + 2x * x + 2x * 2).

### What is the trinomial obtained after simplifying the equation 2x^2 + 4x - 16 = 0?

-The simplified trinomial is x^2 + 2x - 8.

### How does the script factor the trinomial x^2 + 2x - 8?

-The script factors it into (x + 4)(x - 2) by finding numbers that multiply to -8 and add to 2.

### What are the two possible solutions for x in the equation x^2 - 5x + 6 = 0 as per the script?

-The two possible solutions for x are 2 and 3.

### How does the script solve the system of equations x^2 + y^2 = 45 and x^2 - y^2 = -27 using elimination?

-By adding the two equations to get 2x^2 = 18, then dividing by 2 to find x^2 = 9, and taking the square root to get x = Β±3.

### What are the four possible solutions for the system x^2 + y^2 = 45 and x^2 - y^2 = -27?

-The four possible solutions are (3, 6), (-3, 6), (3, -6), and (-3, -6).

###### Outlines

##### π Verifying Solutions in a System of Equations

This paragraph discusses the process of verifying the correctness of solutions in a system of equations. It begins with an example where two possible solutions are checked against two given equations. The first solution (x = -2, y = 5) satisfies both equations, while the second solution (x = 1, y = 2) also meets the criteria, demonstrating that not all potential solutions may be correct. The speaker then moves on to a nonlinear system of equations, using elimination to find a solution (x = 1, y = 3), which is verified to satisfy both original equations. The importance of checking each solution is emphasized.

##### π Solving Nonlinear Equations Using Elimination and Substitution

The second paragraph focuses on solving a system of nonlinear equations using both elimination and substitution methods. The speaker first uses elimination to solve for x and y in an example where the equations are y = x^2 + 3x - 1 and y = x^2 - 4x + 6. After finding x = 1, the value of y is determined to be 3, which satisfies both equations. The speaker then introduces a new example with the equations x^2 + y^2 = 20 and x - y = -2, recommending substitution due to the inability to use elimination. The process involves isolating y and substituting it into the first equation, leading to a quadratic equation in x, which is then solved to find the values of x and y.

##### π Factoring and Solving Quadratic Equations

In this paragraph, the speaker tackles a quadratic equation derived from substituting y = x + 2 into the equation x^2 + (x + 2)^2 = 20. After simplifying, the equation becomes 2x^2 + 4x - 16 = 0, which is then factored into (x + 4)(x - 2) = 0, yielding solutions x = -4 and x = 2. Corresponding y values are found using the equation y = x + 2, resulting in two potential solutions: (-4, -2) and (2, 4). These are verified by plugging them back into the original equations, confirming their correctness.

##### π Solving Systems of Equations with Multiplication and Isolation

The speaker presents a system of equations involving multiplication and isolation: xy = 12 and 2x + y = 10. The recommended method is substitution, starting by isolating y as y = -2x + 10. This expression is then substituted into the first equation to form a quadratic equation in x: -2x^2 + 10x = 12. After simplifying, the equation becomes x^2 - 5x + 6 = 0, which factors into (x - 2)(x - 3) = 0, giving x values of 2 and 3. Substituting these into the equation for y yields the solutions (2, 6) and (3, 4). The speaker suggests verifying these solutions by plugging them back into the original equations.

##### π Solving Nonlinear Equations with Addition and Square Roots

The final paragraph addresses a system of nonlinear equations: x^2 + y^2 = 45 and x^2 - y^2 = -27. The speaker uses addition to eliminate y^2, resulting in 2x^2 = 18, which simplifies to x^2 = 9. The square roots of 9 are Β±3, giving two possible x values. For y, the equation y^2 = 45 - x^2 is used, which simplifies to y^2 = 36 for both x = 3 and x = -3, yielding y values of Β±6. This results in four possible solutions: (3, 6), (-3, 6), (3, -6), and (-3, -6). The speaker explains that all these solutions are valid as they satisfy both original equations, concluding the lesson on solving systems of nonlinear equations.

###### Mindmap

###### Keywords

##### π‘Non-linear equations

##### π‘Substitution

##### π‘Elimination

##### π‘Solution

##### π‘Quadratic equation

##### π‘Factoring

##### π‘Zero product property

##### π‘Checking solutions

##### π‘Squaring

##### π‘System of equations

###### Highlights

Checking multiple solutions for correctness in solving systems of equations.

Using substitution to verify solutions by plugging values into equations.

Demonstration of solving a system of nonlinear equations using elimination method.

Explanation of how to multiply an equation by -1 to facilitate elimination.

Solving for x and then y using the elimination method for a nonlinear system.

Verification of the solution by substituting back into the original equations.

Introduction of a system of equations involving squares of variables.

Recommendation to use substitution over elimination for certain systems.

Process of isolating y and substituting it into the other equation.

Expanding binomials to form a quadratic equation.

Solving the quadratic equation by factoring and using the zero product property.

Determining possible x and y values from the factored equation.

Verification of all possible solutions in the original system of equations.

Approach to solving a system with multiplication and addition of variables.

Using substitution to express y in terms of x and simplify the system.

Factoring a quadratic equation to find the values of x.

Calculating corresponding y values for each x solution.

Quick check method for verifying solutions without full substitution.

Introduction of a system of equations with squared terms requiring elimination.

Combining equations to eliminate variables and simplify the system.

Finding x values by solving the simplified equation.

Determining y values based on the found x values and original equations.

Explanation of why all combinations of x and y values are valid solutions.

Conclusion on solving systems of non-linear equations with various methods.

###### Transcripts

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