You are viewing the article How to Find Intersection Algebraically at Thptlaihoa.edu.vn you can quickly access the necessary information in the table of contents of the article below.
Finding the intersection algebraically between two equations or expressions is an essential skill in algebra. It allows us to determine the common solutions or points of intersection between two lines or curves. By using algebraic methods such as substitution or elimination, we can find precise and accurate coordinates that represent the intersection points. Whether you are solving linear or quadratic equations, understanding how to find intersections algebraically is crucial for various mathematical applications, such as solving systems of equations or analyzing graphical representations. In this guide, we will delve into different algebraic strategies and techniques to find intersections and effectively determine the solution set of given equations or expressions.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 16 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 63,431 times.
When two lines intersect on a two-dimensional coordinate system, they meet only at a point represented by the x and y coordinate pair. Since both lines pass through that point, the x, y coordinate pair must satisfy both equations. With some additional techniques, you can find the intersection of the parabola and other quadratic curves by analogous reasoning.
Steps
Find the intersection of two lines
![Image titled Algebraically Find the Intersection of Two Lines Step 1](https://www.wikihow.com/images/thumb/e/e9/Algebraically-Find-the-Intersection-of-Two-Lines-Step-1-Version-3.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-1-Version-3.jpg)
- If the problem doesn’t say equations, find them from the information you already have.
- Example: Two lines whose equation is y=x+3{displaystyle y=x+3} and y−twelfth=−2x{displaystyle y-12=-2x} . In the second equation, to leave only y on the left side, you add 12 to both sides: y=twelfth−2x{displaystyle y=12-2x}
![Image titled Algebraically Find the Intersection of Two Lines Step 2](https://www.wikihow.com/images/thumb/f/f0/Algebraically-Find-the-Intersection-of-Two-Lines-Step-2-Version-3.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-2-Version-3.jpg)
- Example: We know y=x+3{displaystyle y=x+3} and y=twelfth−2x{displaystyle y=12-2x} , therefore x+3=twelfth−2x{displaystyle x+3=12-2x} .
![Image titled Algebraically Find the Intersection of Two Lines Step 3](https://www.wikihow.com/images/thumb/f/ff/Algebraically-Find-the-Intersection-of-Two-Lines-Step-3-Version-3.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-3-Version-3.jpg)
- For example:x+3=twelfth−2x{displaystyle x+3=12-2x}
- Add 2x{displaystyle 2x} on two sides:
- 3x+3=twelfth{displaystyle 3x+3=12}
- Subtract 3 from both sides:
- 3x=9{displaystyle 3x=9}
- Divide both sides by 3:
- x=3{displaystyle x=3} .
![Image titled Algebraically Find the Intersection of Two Lines Step 4](https://www.wikihow.com/images/thumb/8/83/Algebraically-Find-the-Intersection-of-Two-Lines-Step-4-Version-3.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-4-Version-3.jpg)
- For example:x=3{displaystyle x=3} and y=x+3{displaystyle y=x+3}
- y=3+3{displaystyle y=3+3}
- y=6{displaystyle y=6}
![Image titled Algebraically Find the Intersection of Two Lines Step 5](https://www.wikihow.com/images/thumb/e/e7/Algebraically-Find-the-Intersection-of-Two-Lines-Step-5-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-5-Version-2.jpg)
- For example:x=3{displaystyle x=3} and y=twelfth−2x{displaystyle y=12-2x}
- y=twelfth−2(3){displaystyle y=12-2(3)}
- y=twelfth−6{displaystyle y=12-6}
- y=6{displaystyle y=6}
- Thus we get the same y value. The essay has no errors.
![Image titled Algebraically Find the Intersection of Two Lines Step 6](https://www.wikihow.com/images/thumb/5/5c/Algebraically-Find-the-Intersection-of-Two-Lines-Step-6-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-6-Version-2.jpg)
- For example:x=3{displaystyle x=3} and y=6{displaystyle y=6}
- The two lines intersect at (3,6).
![Image titled Algebraically Find the Intersection of Two Lines Step 7](https://www.wikihow.com/images/thumb/1/19/Algebraically-Find-the-Intersection-of-Two-Lines-Step-7-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-7-Version-2.jpg)
- If the two lines are parallel, then they do not intersect. The x terms are canceled out and the equation is simplified to a false statement (e.g. 0=first{displaystyle 0=1} ). Write the answer as ” two lines do not intersect ” or ” no real solution “.
- If two equations represent the same line, they “intersect” at every point. The x terms are canceled out and the equation is simplified to a true statement (e.g. 3=3{displaystyle 3=3} ). Write the answer as ” two lines coincide “.
Problems with quadratic equations
![Image titled Algebraically Find the Intersection of Two Lines Step 8](https://www.wikihow.com/images/thumb/1/1b/Algebraically-Find-the-Intersection-of-Two-Lines-Step-8-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-8-Version-2.jpg)
- Expand the equations from parentheses to check if they have quadratic form. For example, y=(x+3)(x){displaystyle y=(x+3)(x)} has quadratic form because it is expanded to y=x2+3x.{displaystyle y=x^{2}+3x.}
- Equations of circles and ellipses have both terms x2{displaystyle x^{2}} and y2{displaystyle y^{2}} . [1] XResearch Sources[2] XResearch Resources If you are having trouble with these special cases see the Advice below.
![Image titled Algebraically Find the Intersection of Two Lines Step 9](https://www.wikihow.com/images/thumb/9/9c/Algebraically-Find-the-Intersection-of-Two-Lines-Step-9-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-9-Version-2.jpg)
- Example: Find the intersection of x2+2x−y=−first{displaystyle x^{2}+2x-y=-1} and y=x+7{displaystyle y=x+7} .
- Rewrite the quadratic equation in terms of y:
- y=x2+2x+first{displaystyle y=x^{2}+2x+1} and y=x+7{displaystyle y=x+7} .
- This example has a quadratic equation and a linear equation. Problems with two quadratic equations are solved similarly.
![Image titled Algebraically Find the Intersection of Two Lines Step 10](https://www.wikihow.com/images/thumb/f/f0/Algebraically-Find-the-Intersection-of-Two-Lines-Step-10-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-10-Version-2.jpg)
- For example:y=x2+2x+first{displaystyle y=x^{2}+2x+1} and y=x+7{displaystyle y=x+7}
- x2+2x+first=x+7{displaystyle x^{2}+2x+1=x+7}
![Image titled Algebraically Find the Intersection of Two Lines Step 11](https://www.wikihow.com/images/thumb/3/3a/Algebraically-Find-the-Intersection-of-Two-Lines-Step-11-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-11-Version-2.jpg)
- For example:x2+2x+first=x+7{displaystyle x^{2}+2x+1=x+7}
- Subtract x from both sides:
- x2+x+first=7{displaystyle x^{2}+x+1=7}
- Subtract 7 from both sides:
- x2+x−6=0{displaystyle x^{2}+x-6=0}
![Image titled Algebraically Find the Intersection of Two Lines Step 12](https://www.wikihow.com/images/thumb/a/a1/Algebraically-Find-the-Intersection-of-Two-Lines-Step-12-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-12-Version-2.jpg)
- For example:x2+x−6=0{displaystyle x^{2}+x-6=0}
- The purpose of factoring is to find two factors that, when multiplied together, form an equation. Starting with the first term, we know x2{displaystyle x^{2}} can be decomposed into x and x. Write it in the form (x )(x ) = 0.
- The final term is -6. List each pair of factors that, when multiplied together, equal -6: −6∗first{displaystyle -6*1} , −3∗2{displaystyle -3*2} , −2∗3{displaystyle -2*3} , and −first∗6{displaystyle -1*6} .
- The middle term is x (which can be written as 1x). Add each pair of factors together until you get 1. The correct pair of factors is −2∗3{displaystyle -2*3} , because −2+3=first{displaystyle -2+3=1} .
- Fill in the blanks with this pair of factors: (x−2)(x+3)=0{displaystyle (x-2)(x+3)=0} .
![Image titled Algebraically Find the Intersection of Two Lines Step 13](https://www.wikihow.com/images/thumb/6/6e/Algebraically-Find-the-Intersection-of-Two-Lines-Step-13-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-13-Version-2.jpg)
- Example (factoring): Finally we have the equation (x−2)(x+3)=0{displaystyle (x-2)(x+3)=0} . If either factor is 0, then the equation is satisfied. One solution is x−2=0{displaystyle x-2=0} → x=2{displaystyle x=2} . The remaining solution is x+3=0{displaystyle x+3=0} → x=−3{displaystyle x=-3} .
- Example (quadratic solution formula or square’s complement): If you use either of these two ways to solve the equation, the square root sign will appear. For example, the equation becomes x=(−first+25)/2{displaystyle x=(-1+{sqrt {25}})/2} . Remember that square roots can be simplified to two different solutions: 25=5∗5{displaystyle {sqrt {25}}=5*5} , and25=(−5)∗(−5){displaystyle {sqrt {25}}=(-5)*(-5)} . Write two equations for each case and solve for x respectively.
![Image titled Algebraically Find the Intersection of Two Lines Step 14](https://www.wikihow.com/images/thumb/f/fb/Algebraically-Find-the-Intersection-of-Two-Lines-Step-14-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-14-Version-2.jpg)
- One solution: The problem can be factored into two identical factors ((x-1)(x-1) = 0). When substituting in the quadratic formula, the term whose root is 0{displaystyle {sqrt {0}}} . You only need to solve one equation.
- No real solution: No factor can satisfy the requirement (sum equals the middle term). When you substitute in the square root formula, you get a negative number below the square root sign (for example, −2{displaystyle {sqrt {-2}}} ). Write the answer as “no solution”.
![Image titled Algebraically Find the Intersection of Two Lines Step 15](https://www.wikihow.com/images/thumb/c/ce/Algebraically-Find-the-Intersection-of-Two-Lines-Step-15-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-15-Version-2.jpg)
- Example: We find two solutions x=2{displaystyle x=2} and x=−3{displaystyle x=-3} . One of the two lines has the equation y=x+7{displaystyle y=x+7} . Instead y=2+7{displaystyle y=2+7} and y=−3+7{displaystyle y=-3+7} , then solve each equation to find y=9{displaystyle y=9} and y=4{displaystyle y=4} .
![Image titled Algebraically Find the Intersection of Two Lines Step 16](https://www.wikihow.com/images/thumb/2/2c/Algebraically-Find-the-Intersection-of-Two-Lines-Step-16-Version-2.jpg/v4-728px-Algebraically-Find-the-Intersection-of-Two-Lines-Step-16-Version-2.jpg)
- Example: When replacing x=2{displaystyle x=2} in, we have y=9{displaystyle y=9} , so the intersection has coordinates (2, 9) . Doing the same for the second solution will give the coordinates of the remaining intersection as (-3, 4) .
Advice
- The equations of a circle and an ellipse have a term x2{displaystyle x^{2}}and a number of grades y2{displaystyle y^{2}} . To find the intersection of the circle and the line, solve for x in the linear equation. Substitute the solution for x in the equation of the circle and you will have a quadratic equation that is easier to solve. These problems can have 0, 1, or 2 solutions, as described in the method above.
- The circle and the parabp (or other quadratic) can have 0, 1, 2, 3, or 4 solutions. Find the variable to the power of 2 in both equations — say x 2 . Solve find x2{displaystyle x^{2}} and replace the answer in x2{displaystyle x^{2}} in the remaining equation. Solve for y to get 0, 1 or 2 solutions. Substitute each solution back into the original quadratic equation to solve for x. Each of these equations can have 0, 1, or 2 solutions.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 16 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 63,431 times.
When two lines intersect on a two-dimensional coordinate system, they meet only at a point represented by the x and y coordinate pair. Since both lines pass through that point, the x, y coordinate pair must satisfy both equations. With some additional techniques, you can find the intersection of the parabola and other quadratic curves by analogous reasoning.
In conclusion, finding the intersection algebraically is a straightforward process that involves solving equations simultaneously. By setting two or more equations equal to each other and solving for the variables, we can find the coordinates of the points where the graphs intersect. This method is particularly useful when trying to find the solution to a system of equations or when graphing linear or quadratic functions. Additionally, algebraic methods allow us to precisely determine the coordinates of the intersection points, even when they are not easily visible on a graph. However, it is important to note that this method can become more complex as the number of equations and variables increases. In such cases, it may be helpful to utilize technology or employ alternative solution techniques. Overall, algebraic methods provide a powerful tool for finding intersections and can be widely applied in various mathematical contexts.
Thank you for reading this post How to Find Intersection Algebraically at Thptlaihoa.edu.vn You can comment, see more related articles below and hope to help you with interesting information.
Related Search:
1. Algebraic methods for finding the intersection of two lines
2. Solving systems of equations algebraically to find the intersection
3. Using algebra to find the intersection point of a line and a parabola
4. Algebraic approach to finding the intersection of two quadratic equations
5. Steps to algebraically determine the intersection of a line and a circle
6. Algebraic methods for finding the intersection of two exponential functions
7. Using algebra to find the intersection point of two trigonometric functions
8. Algebraic techniques for finding the intersection point of two logarithmic equations
9. Steps to algebraically determine the intersection of two rational functions
10. Algebraic approaches for finding the intersection of a polynomial and a radical function