A Point Where Two or More Functions Intersect.

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When straight lines intersect on a two-dimensional graph, they meet at only one point,^[1] described by a single set of ${\displaystyle x}$ - and ${\displaystyle y}$ -coordinates. Because both lines pass through that point, you know that the ${\displaystyle x}$ - and ${\displaystyle y}$ - coordinates must satisfy both equations. With a couple extra techniques, you can find the intersections of parabolas and other quadratic curves using similar logic.

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   Check your work. It's a good idea to plug your ${\displaystyle x}$ -value into the other equation and see if you get the same result. If you get a different solution for ${\displaystyle y}$ , go back and check your work for mistakes.^[6]

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   Deal with unusual results. Some equations make it impossible to solve for ${\displaystyle x}$ . This doesn't always mean you made a mistake. There are two ways a pair of lines can lead to a special solution:

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   Recognize quadratic equations. In a quadratic equation, one or more variables is squared ( ${\displaystyle x^{2}}$ or ${\displaystyle y^{2}}$ ), and there are no higher powers. The lines these equations represent are curved, so they can intersect a straight line at 0, 1, or 2 points. This section will teach you how to find the 0, 1, or 2 solutions to your problem.

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   Write the equations in terms of y. If necessary, rewrite each equation so y is alone on one side.

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   Combine the two equations to cancel out the y. Once you've set both equations equal to y, you know the two sides without a y are equal to each other.

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   Arrange the new equation so one side is equal to zero. Use standard algebraic techniques to get all the terms on one side. This will set the problem up so we can solve it in the next step.

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   Keep an eye out for two solutions for x. If you work too quickly, you might find one solution to the problem and not realize there's a second one. Here's how to find the two x-values for lines that intersect at two points:

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   Solve problems with one or zero solutions. Two lines that barely touch only have one intersection, and two lines that never touch have zero. Here's how to recognize these:

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   Plug your x-values back into either original equation. Once you have the x-value of your intersection, plug it back into one of the equations you started with. Solve for y to find the y-value. If you have a second x-value, repeat for this as well.

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   Write the point coordinates. Now write your answer in coordinate form, with the x-value and y-value of the intersection points. If you have two answers, make sure you match the correct x-value to each y-value.

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• Question

  What happens if the x's cancel out?

  

  Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

  

  Assistant Professor of Mathematics

  Expert Answer

• Question

  F(x)=2^2=12x+10 , g(x)=38

  

  I suspect that you copied this problem down wrong. I'll deal with what you wrote first, and then I'll talk about what I think you may have meant. As written, the first function says F(x)=2^2=12x+10. In other words, this is a simple one variable equation that simplifies to 4=12x+10. Then subtract 10 from both sides to get -6=12x. Finally, divide both sides by 12 to get -1/2 = x. You now have two different functions, each with a single variable. F(x): x=-1/2, and G(x): x=38. Any function that has only a single variable like this, x=__, is going to be a vertical straight line at that value. As a result, these two lines will never intersect, and there is no single solution for F(x) and G(x) simultaneously. That is not a very interesting solution, which makes me think you copied it wrong. I think that what you probably meant is F(x)=x^2 + 12x + 10. I think you wrote 2^2 instead of x^2, and then you changed a + symbol into an = symbol in the middle of the function. (The + and = are the same button on most keyboards.) This becomes a more interesting problem. You could now work on factoring the first function, but you don't need to do that much work. If you notice, the second function, G(x), is already solved. It is the single value, G(x)=38. This means that the graph of that function is a straight vertical line. At every point on the line, x=38. So to solve the system, just insert the value 38 for x in the first equation: F(x)=38^2+12(38)+10. This equals 1444+456+10, which is F(x)=1910. So the solution where those two graphs cross is x=38, y=1910. You can write the coordinate pair as (38,1910).

• Question

  When the lines intersect at (3,6), what could represent the two lines?

  

  The lines could be x = 3 and y = 6.

• Question

  How do I get the points of intersection of two equations on a straight line?

  

  If you are asking about two linear (straight line) equations, there will be only one point of intersection. This is explained in Method 1 above.

• Question

  I have 2 lines that intersect. I know only slope of the lines and one Y value of each line at the same unknown x. How do I find intersection point?

  

  Because the x value of the specified points is unknown, you don't know where the specified points lie, and therefore you can't find either the y-intercept of the lines or their slope-intercept equations. Therefore, you cannot determine the point of intersection.

• Question

  What if the equation doesn't factor out?

  

  Technist

  Community Answer

  Remember that factoring only works with quadratic equations. If completing the square doesn't work, try using the quadratic equation and vice versa.

• Question

  What if there isn't an isolated variable? For example, 4x + 10y = 5 and 5x + 8y = 5

  

  Isolate either variable yourself. For example, in the first equation, isolate and solve for x by subtracting 10y from both sides and then dividing both sides by 4. Isolate and solve for y by subtracting 4x from both sides and then dividing both sides by 10.

• Question

  What are the intersect points for x + y = -3 and -x + y = 3?

  

  Because each equation represents a straight line, there will be just one point of intersection. The easiest way to solve for x and y is to add the two equations together (by adding the left sides together, adding the right sides together, and setting the two sums equal to each other): (x+y) + (-x+y) = (-3) + (3). Then 2y = 0, and y = 0. Substitute the y value into either of the original equations to find the x value: x + 0 = -3, and x = -3. So the point of intersection of the two lines is (-3,0).

• Question

  How do I find the line that passes through the point of intersection and a perpendicular line?

  

  Use the quadratic equation -b(square root) b^2-4ac / 2a.

• Question

  What is the inter sect of these two? y=-0.1x^{2}\ +x+4 and y=0.2x+1?

  

  The intersection occurs at the point(s) where the two equations equal each other. So set one equation equal to the other, and solve for x. Then substitute that x value back into either equation to get the y value. You then have the x and y values of the point of intersection.

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• Equations for a circle or ellipse have an ${\displaystyle x^{2}}$ term and a ${\displaystyle y^{2}}$ term. To find the intersection of a circle and a straight line, solve for x in the linear equation.^[10] Substitute the solution for x in the circle equation, and you'll have an easier quadratic equation. These problems can have 0, 1, or 2 solutions, as described in the method above.

• A circle and a parabola (or other quadratic) can have 0, 1, 2, 3, or 4 solutions. Find the variable that is squared in both equations — let's say it's x². Solve for ${\displaystyle x^{2}}$ and substitute the answer for the ${\displaystyle x^{2}}$ in the other equation. Solve for y to get 0, 1, or 2 solutions. Plug each solution back into the original quadratic equation and solve for x. Each of these can have 0, 1, or 2 solutions.

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Article Summary X

To algebraically find the intersection of two straight lines, write the equation for each line with y on the left side. Next, write down the right sides of the equation so that they are equal to each other and solve for x. Write down one of the two equations again, substituting the previous answer in place of x, and solve for y. These answers will give you the x and y coordinates of the intersection. To learn how to find the intersection when working with quadratic equations, keep reading!

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A Point Where Two or More Functions Intersect.

Source: https://www.wikihow.com/Algebraically-Find-the-Intersection-of-Two-Lines

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