How to Find the Intersection of Two Lines (With Sample Problems)

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Associate Professor of Mathematics Mario Bañuelos, PhD, shares simple steps for finding the point of intersection

Co-authored by Mario Banuelos, PhD and Sophie Burkholder, BA

Last Updated: December 29, 2025 Fact Checked

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• For Two Straight Lines
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• For One or More Curved Lines
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• Examples
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• Is there a formula?
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• More Practice Problems
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• Video
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• Expert Q&A

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Working on your algebra homework and needing to find the intersection of two lines, but don’t know where to start? We’ve got you! Below, you’ll find clear and simple steps on finding a point of intersection between two straight lines or a straight and quadratic line. Plus, we’ll share some sample practice problems, as well as step-by-step guidance from Mario Bañuelos, PhD, an Associate Professor of Mathematics.

How do you find the intersection of two lines?

Associate Professor of Mathematics Mario Bañuelos, PhD, says to start with your two lines in $y=mx+b$ format. Then, follow these steps:

1. Set the equations equal to each other (e.g., $mx + b = mx + b$).
2. Use algebraic steps to solve for $x$ (e.g., $mx - mx = b - b$).
3. Once you have $x$ plug it into one equation and solve for $y$.
4. Check your work by plugging $x$ into the other equation.
5. If you don’t get the same answer for $y$, retrace your steps to find your error.
6. Write your final answer in coordinate format: i.e., $(x,y)$.

Steps

Section 1 of 4:

How to Find the Intersection of Two Straight Lines

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1. 
   1
   Write both equations so that they each have $$ y {\displaystyle y} on the left side. Hopefully, both of your equations will already be written so that $y$ is alone on one side of the equals sign, and both your equations are written in this format: $y=mx+b$. However, if this is not the case, you’ll have to adjust one or both equations to fit this format, says Dr. Bañuelos. Remember that, to change the format of an equation, you can cancel out or add terms by performing the same action to both sides.

   • For example, let’s say that your two lines have the equations $y=x+3$ and $y-12=-2x$.
     • The first equation is the correct format. In the second equation, however, we need to isolate $y$.
     • The easiest way to do this is to add $12$ to both sides of the equation. Then, we’ll end up with $y=12-2x$ as our second line.
   • What if my equations are in point-slope form? If one of your lines is in point-slope form (i.e., y - y₁ = m(x - x₁)), you’ll need to convert it into $y=mx+b$ before moving forward.
     • To do this, first distribute your slope ($m$) to get y - y₁ = mx - mx₁.
     • Then, add y - y₁ to both sides to get y = mx - mx₁ + y₁. Now, your equation is in the proper format!
   • What if one line is vertical? If one line is vertical, you’ll know because the equation will have no slope or variable. It might look like, for instance, $x=12$. If you have an equation like this, you already have the value for $x$ in your point of intersection! So, all you need to do is plug your $x$ value into your other equation and solve for $y$. Then, you’ll have both coordinates for your point of intersection!

   Meet the wikiHow expert

   Mario Bañuelos, PhD, is an Associate Professor of Mathematics at California State University, Fresno. He holds a BA in Mathematics from CSU-Fresno, and a Ph.D. in Applied Mathematics from UC-Merced.
2. 
   2
   Set the right sides of the equation equal to each other. Now that we have $y$ on the left side of both equations, we know that the right sides of each equation are equal to each other (this is because the lines cross where they have the same $x$ and $y$ values). Write a new equation where both lines are equal to one another, instructs Dr. Bañuelos. In other words, take the non-$y$ sides of each equation and put an equals sign between them.

   • For example, if you want to know where the lines $y=x+3$ crosses $y=12-2x$, you'd equate them by writing $x+3=12-2x$.
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3. 
   3
   Solve for $$ x {\displaystyle x} . The new equation only has one variable: $x$. Use algebra to solve for $x$ by performing the same operation (or operations) on both sides. Continue until you have $x$ alone on one side of the equation, instructs Dr. Bañuelos.

   • For example, these are the steps to solve for $x$ in $x+3=12-2x$:
     • Add $2x$ to each side: $3x+3=12$.
     • Subtract $3$ from each side: $3x=9$.
     • Divide each side by $3$: $x=3$.
   • If it’s impossible to solve for $x$ in your equation, skip to the last step in this section for tips and tricks.
4. 
   4
   Use your new $$ x {\displaystyle x} value to solve for $$ y {\displaystyle y} . Now that you know what $x$ is equal to, Dr. Bañuelos says to replace every $x$ in your first equation with your numerical answer. Do the arithmetic to solve for $y$.

   • For example, our first equation was $y=x+3$ and our $x$ value was $3$.
   • Therefore, we’d plug $3$ into $y=x+3$ to get $y=3+3$.
   • Using arithmetic, we can solve and simplify to get $y=6$.
5. 
   5
   Check your work by plugging $$ x {\displaystyle x} into your second equation. It’s a good idea to make sure that your answers are correct by plugging your $x$ value into your other equation to make sure that you get the same result. If you get a different solution for $y$, go back and check your work for mistakes.

   • For example, our second equation was $y=12-2x$ and our $x$ value was $3$.
   • Therefore, we’d plug $3$ into $y=12-2x$ to get $y=12-2(3)$.
   • Using arithmetic, we can solve and simplify to get $y=12-6$.
   • We’ll then simplify further to get $y=6$.
   • Since we also got $6$ in the previous step, we know that our answer is correct!
6. 
   6
   Write down your $$ x {\displaystyle x} and $$ y {\displaystyle y} values as a coordinate pair. To properly identify the point of intersection, we need to write the point’s $x$ and $y$ values as coordinates. So, explains Dr. Bañuelos, write down $x$ and $y$ as a coordinate pair, following this format: $(x,y)$.

   • For example, our values are $x=3$ and $y=6$.
   • The two lines intersect at $(3,6)$.
7. 
   7
   Troubleshoot any unusual results if necessary. Some equations will make it impossible to solve for $x$, but this doesn’t always mean you’ve made a mistake. There are two ways that a pair of lines could lead to an unusual solution.

   • If the lines are parallel to each other, they will never intersect. The $x$ terms will cancel out, and your equation will simplify to a false statement (such as $0=1$), according to Dr. Bañuelos. In this case, write "the lines do not intersect" or no real solution" as your answer.
   • If the two equations describe the same line, they will intersect everywhere. Algebraically, the $x$ terms will cancel out, and your equation will simplify to a true statement (such as $3=3$). In this case, write "the two lines are the same" as your answer.
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Section 2 of 4:

How to Find the Intersection of Curved Lines

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1. 
   1
   Know how to spot a curved line equation (AKA, a quadratic equation). In a quadratic equation, one or more variables are squared ($x^{2}$ or $y^{2}$), 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. The following steps in this section will teach you how to find the 0, 1, or 2 solutions to your problem.

   • If your equation has parentheses in it, expand it to identify whether or not it’s quadratic. For example, $y=(x+3)(x)$ is quadratic, since it expands into $y = x^2 + 3x$.
   • If your equation is a circle or ellipse, it’ll have both an $x^{2}$ and a $y^{2}$ term.
2. 
   2
   Write both equations so that $$ y {\displaystyle y} is alone on one side. If your equations aren’t already set up this way, you can make them so by performing the same actions on both sides of a single equation.

   • For example, let’s say we want to find the intersection of $x^{2}+2x-y=-1$ and $y=x+7$:
     • Rewrite the quadratic equation in terms of $y$, to end up with $y=x^{2}+2x+1$ and $y=x+7$.
     • This example has one quadratic equation and one linear equation. Problems with two quadratic equations are solved in a similar way
3. 
   3
   Combine the two equations and set them equal to each other to cancel out $$ y {\displaystyle y} . Once you've set both equations equal to $y$, you know the two sides without a $y$ are equal to each other. Take the non-$y$ sides of each equation and put an equals sign between them to create one equation, instructs Dr. Bañuelos.

   • For example, with $y=x^{2}+2x+1$ and $y=x+7$, we’ll set them equal to each other to end up with $x^{2}+2x+1=x+7$.
4. 
   4
   Arrange the new equation so that one side is equal to zero. Use standard algebraic techniques to get all the terms on one side, leaving the other side with zero. This will set the problem up so we can solve it in the next step.

   • As an example, we’re starting with this equation: $x^{2}+2x+1=x+7$.
     • Subtract $x$ from each side: $x^{2}+x+1=7$.
     • Subtract $7$ from each side: $x^{2}+x-6=0$.
5. 
   5
   Solve the quadratic equation through factoring (or your preferred method). Once you've set one side equal to zero, there are three ways to solve a quadratic equation. You can use the quadratic formula, complete the square, or use the factoring method. You should use whatever method feels easiest to you. If you’re unsure, we will present you with the steps to the factoring method below:

   • Recall that the goal of factoring is to find the two factors that multiply together to make this equation.
   • Our example equation is $x^{2}+x-6=0$.
     • First, we know that $x^{2}$ can divide into $x$ and $x$. Write down $(x )(x ) = 0$ to show this.
     • The last term is $-6$. List each pair of factors that multiply to make negative six: $-6*1$, $-3*2$, $-2*3$, and $-1*6$.
     • The middle term is $x$ (which you could write as $1x$). Add each pair of factors together until you get 1 as an answer. The correct pair of factors is $-2*3$, since $-2+3=1$.
     • Fill out the gaps in your answer with this pair of factors: $(x-2)(x+3)=0$.
6. 
   6
   Consider whether there might be a second solution for $$ x {\displaystyle x} . Remember that you’re working with a curved line, so it’s possible to get 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 continue your factoring steps in order to find the two x-values for lines that intersect at two points:

   • Continuing our example, we ended up with the equation $(x-2)(x+3)=0$. If either of the factors in parentheses equals $0$, the equation is true.
     • One solution is $x-2=0$ → $x=2$. The other solution is $x+3=0$ → $x=-3$.
   • If you used the quadratic equation or completed the square to solve your equation, a square root will show up in the case of two points of intersection.
     • For example, our equation becomes $x=(-1+{\sqrt {25}})/2$. Remember that a square root can simplify to two different solutions: ${\sqrt {25}}=5*5$, and ${\sqrt {25}}=(-5)*(-5)$.
     • Write two equations, one for each possibility, and solve for x in each one.
7. 
   7
   Recognize when there is only one or zero solutions for $$ x {\displaystyle x} . Two lines that barely touch only have one intersection, and two lines that never touch have zero. Here's how to recognize these:

   • Is there only one solution? The problems factor into two identical factors $((x-1)(x-1) = 0)$. When plugged into the quadratic formula, the square root term is ${\sqrt {0}}$. You only need to solve one equation.
   • Are there zero real solutions? There are no factors that satisfy the requirements (summing to the middle term). When plugged into the quadratic formula, you get a negative number under the square root sign (such as ${\sqrt {-2}}$). Write "no solution" as your answer.
8. 
   8
   Plug your x-values back into either original equation to find y. 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.

   • For example, we found two solutions: $x=2$ and $x=-3$.
   • One of our lines has the equation $y=x+7$. Plug in $y=2+7$ and $y=-3+7$, then solve each equation to find that $y=9$ and $y=4$.
9. 
   9
   Write your answers in coordinate form. Record your results 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.

   • In our example, when we plugged in $x=2$, we got $y=9$, so one intersection is at $(2,9)$. The same process for our second solution tells us another intersection lies at $(-3, 4)$. Those coordinates are our final answers!
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Section 3 of 4:

Example Problems & Solutions

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1. 1
   Example 1 Find the point of intersection for $y=3x-3$ and $y = 2x + 4$:

   • Solution:
     • Both equations already have $y$ isolated on the lefthand side.
     • Set the equations equal to each other: $3x - 3 = 2x + 4$.
     • Solve for $x$: $3x - 2x = 7$ → $x=7$.
     • Use the x-value to solve for $y$: $y = 3(7) - 3 = 18$.
     • Check your work: $y = 2(7) + 4 = 18$.
     • Record your answer: $(7, 18)$
2. 2
   Example 2 Find the point of intersection for $y = 10 - x$ and $y = 4x - 5$:

   • Solution:
     • Both equations already have $y$ isolated on the lefthand side.
     • Set the equations equal to each other: $10 - x = 4x - 5$.
     • Solve for $x$: $10 + 5 = 4x + x$ → $15 = 5x$ → $3=x$.
     • Use the x-value to solve for $y$: $y = 4(3) - 5 = 7$.
     • Check your work: $y = 10 - 3 = 7$.
     • Record your answer: $(3, 7)$
3. 3
   Example 3 Find the point of intersection for $2y + 6 = 4x$ and $y = x + 2$:

   • Solution:
     • Set the first equation so that $y$ is isolated: $2y + 6 = 4x$ → $2y = 4x - 6$ → $y = 2x - 3$.
     • Set the equations equal to each other: $2x - 3 = x + 2$.
     • Solve for $x$: $2x - 3 = x + 2$ → $2x - x = 2 + 3$ → $x=5$.
     • Use the x-value to solve for $y$: $y = 2(5) - 3 = 7$.
     • Check your work: $y = 5 + 2 = 7$
     • Record your answer: $(5, 7)$
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Section 4 of 4:

Is there a formula for finding a point of intersection?

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1. Yes, there is a direct formula for points of intersection, but it’s complex. While the formula may be helpful for some, most mathematicians, teachers, and students prefer the substitution method, as the formula is rather hefty and complex. However, if you’re interested in using the direct formula, you can find it below:

   • Point of Intersection Direct Formula: (x, y) = ((b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁), (a₂c₁ - a₁c₂)/(a₁b₂ - a₂b₁))
     • Note that in order to use this formula, your equations must be in the form a₁ + b₁y + c₁ = 0.

Practice Problems and Answers

Practice Problems and Answers to Find the Intersection of Two Lines

Expert Q&A

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• Question

  What happens if the x's cancel out?

  

  Mario Banuelos, PhD
  Associate Professor of Mathematics

  Mario Banuelos is an Associate 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.

  

  Mario Banuelos, PhD

  Associate Professor of Mathematics

  Expert Answer

  If that happens, you'll end up with a contradiction (like 1 = 2), which means that those two lines will never intersect.

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  Thank you for your feedback.
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  Not Helpful 1 Helpful 8
• Question

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

  

  Community Answer

  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).

  Thanks! We're glad this was helpful.
  Thank you for your feedback.
  If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow

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• Question

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

  

  Donagan

  Top Answerer

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

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  Thank you for your feedback.
  If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow

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