Content

• Steps
• Method 1 of 2: Intersection point of two lines
• Method 2 of 2: Problems with Quadratic Functions
• Tips

In two-dimensional space, two straight lines intersect only at one point, specified by coordinates (x, y). Since both lines pass through the point of their intersection, the coordinates (x, y) must satisfy both equations that describe these lines.With some additional skill, you can find the intersection points of parabolas and other quadratic curves.

Steps

Method 1 of 2: Intersection point of two lines

1. 1 Write down the equation for each line by isolating the y variable on the left side of the equation. The other terms in the equation should be placed on the right side of the equation. Perhaps the equation given to you will contain the variable f (x) or g (x) instead of "y"; in this case, isolate such a variable. To isolate a variable, perform the appropriate math on both sides of the equation.
   • If the equations of the straight lines are not given to you, find them based on the information you know.
   • Example... Given are straight lines described by the equations ${ displaystyle y = x + 3}$ and ${ displaystyle y-12 = -2x}$... To isolate the y in the second equation, add 12 to both sides of the equation: ${ displaystyle y = 12-2x}$
2. 2 Equate the expressions on the right side of each equation. Our task is to find the point of intersection of both lines, that is, the point whose coordinates (x, y) satisfy both equations. Since the variable "y" is located on the left side of each equation, the expressions located on the right side of each equation can be equated. Write down the new equation.
   • Example... As ${ displaystyle y = x + 3}$ and ${ displaystyle y = 12-2x}$, then you can write the following equality: ${ displaystyle x + 3 = 12-2x}$.
3. 3 Find the value of the variable "x". The new equation contains only one variable "x". To find the "x", isolate this variable on the left side of the equation by performing the appropriate math on both sides of the equation. You should get an equation of the form x = __ (if this is not possible, skip to the end of this section).
   • Example. ${ displaystyle x + 3 = 12-2x}$
   • Add ${ displaystyle 2x}$ to each side of the equation:
   • ${ displaystyle 3x + 3 = 12}$
   • Subtract 3 from each side of the equation:
   • ${ displaystyle 3x = 9}$
   • Divide each side of the equation by 3:
   • ${ displaystyle x = 3}$.
4. 4 Use the found value of the variable "x" to calculate the value of the variable "y". To do this, substitute the found value "x" in the equation (any) straight line.
   • Example. ${ displaystyle x = 3}$ and ${ displaystyle y = x + 3}$
   • ${ displaystyle y = 3 + 3}$
   • ${ displaystyle y = 6}$
5. 5 Check your answer. To do this, substitute the value "x" in another equation of the line and find the value "y". If you get different y values, check that your calculations are correct.
   • Example:${ displaystyle x = 3}$ and ${ displaystyle y = 12-2x}$
   • ${ displaystyle y = 12-2 (3)}$
   • ${ displaystyle y = 12-6}$
   • ${ displaystyle y = 6}$
   • We got the same value for "y", so there are no errors in our calculations.
6. 6 Write down the coordinates (x, y). By calculating the x and y values, you have found the coordinates of the intersection of the two lines. Write down the coordinates of the intersection point in the form (x, y).
   • Example. ${ displaystyle x = 3}$ and ${ displaystyle y = 6}$
   • Thus, two lines intersect at a point with coordinates (3,6).
7. 7 Calculations in special cases. In some cases, the value of the variable "x" cannot be found. But that doesn't mean you've made a mistake. A special case occurs when one of the following conditions is met:
   • If two lines are parallel, they do not intersect. In this case, the variable "x" will simply be canceled, and the equation will turn into a meaningless equality (for example, ${ displaystyle 0 = 1}$). In this case, write in your answer that straight lines do not intersect or no solution.
   • If both equations describe one straight line, then there will be an infinite set of intersection points. In this case, the variable "x" will simply be canceled, and the equation will turn into a strict equality (for example, ${ displaystyle 3 = 3}$). In this case, write in your answer that two straight lines coincide.

Method 2 of 2: Problems with Quadratic Functions

1. 1 Definition of a quadratic function. In a quadratic function, one or more variables have the second degree (but not higher), for example, ${ displaystyle x ^ {2}}$ or ${ displaystyle y ^ {2}}$... Quadratic function plots are curves that may not or intersect at one or two points. In this section, we will show you how to find the point or points of intersection of quadratic curves.
   • If the equation includes an expression in parentheses, expand the parentheses to ensure that the function is quadratic. For example, the function ${ displaystyle y = (x + 3) (x)}$ is quadratic, since expanding the parentheses gives ${ displaystyle y = x ^ {2} + 3x.}$
   • The function describing the circle includes both ${ displaystyle x ^ {2}}$and ${ displaystyle y ^ {2}}$... If you have any problems solving problems with this function, go to the "Tips" section.
2. 2 Rewrite each equation by isolating the y variable on the left side of the equation. The other terms in the equation should be placed on the right side of the equation.
   • Example... Find the point (s) of intersection of the graphs ${ displaystyle x ^ {2} + 2x-y = -1}$ and ${ displaystyle y = x + 7}$
   • Insulate the y variable on the left side of the equation:
   • ${ displaystyle y = x ^ {2} + 2x + 1}$ and ${ displaystyle y = x + 7}$.
   • In this example, you are given one quadratic function and one linear function. Remember that if you are given two quadratic functions, the calculations are similar to the steps below.
3. 3 Equate the expressions on the right side of each equation. Since the variable "y" is located on the left side of each equation, the expressions located on the right side of each equation can be equated.
   • Example. ${ displaystyle y = x ^ {2} + 2x + 1}$ and ${ displaystyle y = x + 7}$
   • ${ displaystyle x ^ {2} + 2x + 1 = x + 7}$
4. 4 Transfer all terms of the resulting equation to its left side, and write 0 on the right side. To do this, perform basic math operations. This will allow you to solve the resulting equation.
   • Example. ${ displaystyle x ^ {2} + 2x + 1 = x + 7}$
   • Subtract "x" from both sides of the equation:
   • ${ displaystyle x ^ {2} + x + 1 = 7}$
   • Subtract 7 from both sides of the equation:
   • ${ displaystyle x ^ {2} + x-6 = 0}$
5. 5 Solve the Quadratic Equation. Moving all the terms of the equation to its left side, you get a quadratic equation. It can be solved in three ways: using a special formula, complementing to a full square, and factoring the equation.
   • Example. ${ displaystyle x ^ {2} + x-6 = 0}$
   • When factoring an equation, you get two binomials that you multiply to get the original equation. In our example, the first term ${ displaystyle x ^ {2}}$ can be expanded into x * x. Make the following entry: (x) (x) = 0
   • In our example, the free term -6 can be expanded into the following factors: ${ displaystyle -6 * 1}$, ${ displaystyle -3 * 2}$, ${ displaystyle -2 * 3}$, ${ displaystyle -1 * 6}$.
   • In our example, the second term is x (or 1x). Add each pair of intercept factors (in our example -6) until you get 1. In our example, the appropriate pair of intercept factors are -2 and 3 (${ displaystyle -2 * 3 = -6}$), as ${ displaystyle -2 + 3 = 1}$.
   • Fill in the blanks with the found pair of numbers: ${ displaystyle (x-2) (x + 3) = 0}$.
6. 6 Don't forget about the second intersection point of the two graphs. In a hurry, you can forget about the second intersection point. Here's how to find the x-coordinates of two intersection points:
   • Example (factorization)... If in the equation ${ displaystyle (x-2) (x + 3) = 0}$ one of the expressions in brackets will be equal to 0, then the whole equation will be equal to 0. Therefore, you can write it like this: ${ displaystyle x-2 = 0}$ → ${ displaystyle x = 2}$ and ${ displaystyle x + 3 = 0}$ → ${ displaystyle x = -3}$ (that is, you found two roots of the equation).
   • Example (using a formula or complement to a full square)... When using one of these methods, the square root will appear in the solution process. For example, the equation from our example will take the form ${ displaystyle x = (- 1 + { sqrt {25}}) / 2}$... Remember, you get two solutions when you take the square root. In our case: ${ displaystyle { sqrt {25}} = 5 * 5}$, and${ displaystyle { sqrt {25}} = (- 5) * (- 5)}$... So write down two equations and find two x values.
7. 7 The graphs intersect at one point or do not intersect at all. Such situations occur when the following conditions are met:
   • If the graphs intersect at one point, then the quadratic equation is decomposed into the same factors, for example, (x-1) (x-1) = 0, and the square root of 0 appears in the formula (${ displaystyle { sqrt {0}}}$). In this case, the equation has only one solution.
   • If the graphs do not intersect at all, then the equation is not decomposed into factors, and the square root of a negative number appears in the formula (for example, ${ displaystyle { sqrt {-2}}}$). In this case, write in the answer that no solution.
8. 8 Substitute the found value of the variable "x" in the equation (any) of the curve. This will find the value of the y variable. If you have two values ​​for the variable "x", follow the described process with both values ​​of "x".
   • Example... You found two values ​​for the variable "x": ${ displaystyle x = 2}$ and ${ displaystyle x = -3}$... Plug each of these values ​​into a linear equation ${ displaystyle y = x + 7}$... You'll get : ${ displaystyle y = 2 + 7 = 9}$ and ${ displaystyle y = -3 + 7 = 4}$.
9. 9 Write down the coordinates of the intersection point in the form (x, y). By calculating the x and y values, you have found the coordinates of the intersection of the two graphs. If you have identified two values ​​"x" and "y", write down the two pairs of coordinates without confusing the corresponding values ​​"x" and "y".
   • Example... When substituted into the equation ${ displaystyle x = 2}$ You'll get ${ displaystyle y = 9}$, that is, one pair of coordinates (2, 9)... By doing the same calculation with the second x-value, you will get the second pair of coordinates (-3, 4).

Tips

• The function describing the circle includes both ${ displaystyle x ^ {2}}$and ${ displaystyle y ^ {2}}$... To find the intersection point (s) of a circle and a straight line, calculate "x" using a linear equation. Then plug the found x value into the function that describes the circle, and you get a simple quadratic equation that may not have a solution or have one or two solutions.
• A circle and a curve (quadratic or otherwise) may not intersect or intersect at one, two, three, four points. In this case, you need to find the value of x (not "x"), and then substitute it into the second function. By calculating y, you get one or two solutions, or no solutions at all. Now plug the found value "y" into one of the two functions and find the value "x". In this case, you will get one or two solutions, or no solutions at all.