Write the coordinates of the point where these two lines as described above intersect

Write the answer of each of the following questions:

(i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?

(ii) What is the name of each part of the plane formed by these two lines?

(iii) Write the name of the point where these two lines intersect.

Answer (i)

Horizontal line is called x-axis and Vertical line is called y-axis.

Answer (ii)

Each part of the Cartesian plane formed by x-axis and y-axis is called quadrant.

Answer (iii)

x-axis and y-axis meet at one point which is called origin.

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When straight lines intersect on a two-dimensional graph, they meet at only one point,[1] X Research source Go to source described by a single set of

1. 1

   Write the equation for each line with on the left side. If necessary, rearrange the equation so is alone on one side of the equal sign. If the equation uses

   
   or
   
   instead of , separate this term instead. Remember, you can cancel out terms by performing the same action to both sides.

2. 2

   Set the right sides of the equation equal to each other. We're looking for a point where the two lines have the same and values; this is where the lines cross. Both equations have just on the left side, so we know the right sides are equal to each other. Write a new equation that represents this.

3. 3

4. 4

   Use this -value to solve for . Choose the equation for either line. Replace every in the equation with the answer you found. Do the arithmetic to solve for .[5] X Expert Source

   
   Mario Banuelos, PhD
   Assistant Professor of Mathematics Expert Interview. 11 December 2021. Go to source

5. 5

   Check your work. It's a good idea to plug your -value into the other equation and see if you get the same result. If you get a different solution for , go back and check your work for mistakes.[6] X Expert Source

   
   Mario Banuelos, PhD
   Assistant Professor of Mathematics Expert Interview. 11 December 2021. Go to source
   • Example:
     
     and
     
   • 
   • 
   • 
   • This is the same answer as before. We did not make any mistakes.

6. 6

   Write down the and coordinates of the intersection. You've now solved for the -value and -value of the point where the two lines intersect. Write down the point as a coordinate pair, with the -value as the first number.[7] X Expert Source

   
   Mario Banuelos, PhD
   Assistant Professor of Mathematics Expert Interview. 11 December 2021. Go to source
   • Example: and
   • The two lines intersect at (3,6).

7. 7

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

   • If the two lines are parallel, they do not intersect. The terms will cancel out, and your equation will simplify to a false statement (such as
     
     ). Write "the lines do not intersect" or no real solution" as your answer.
   • If the two equations describe the same line, they "intersect" everywhere. The terms will cancel out and your equation will simplify to a true statement (such as
     
     ). Write "the two lines are the same" as your answer.

1. 1

   Recognize quadratic equations. In a quadratic equation, one or more variables is squared (

   
   or
   
   ), 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.
   • Expand equations with parentheses to check whether they're quadratics. For example,
     
     is quadratic, since it expands into
     
   • Equations for a circle or ellipse have both an and a term.[8] X Research source Go to source [9] X Research source Go to source If you're having trouble with these special cases, see the Tips section below.

2. 2

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

   • Example: Find the intersection of
     
     and
     
     .
   • Rewrite the quadratic equation in terms of y:
   • 
     and .
   • 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 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.

4. 4

   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.

   • Example:
     
   • Subtract x from each side:
   • 
   • Subtract 7 from each side:
   • 

5. 5

   Solve the quadratic equation. Once you've set one side equal to zero, there are three ways to solve a quadratic equation. Different people find different methods easier. You can read about the quadratic formula or "completing the square", or follow along with this example of the factoring method:

   • Example:
   • The goal of factoring is to find the two factors that multiply together to make this equation. Starting with the first term, we know 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:
     
     ,
     
     ,
     
     , and
     
     .
   • 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 , since
     
     .
   • Fill out the gaps in your answer with this pair of factors:
     
     .

6. 6

   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:

   • Example (factoring): We ended up with the equation . If either of the factors in parentheses equal 0, the equation is true. One solution is
     
     →
     
     . The other solution is
     
     →
     
     .
   • Example (quadratic equation or complete the square): If you used one of these methods to solve your equation, a square root will show up. For example, our equation becomes
     
     . Remember that a square root can simplify to two different solutions:
     
     , and
     
     . Write two equations, one for each possibility, and solve for x in each one.

7. 7

   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:

   • 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
     
     . You only need to solve one equation.
   • No real solution: 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
     
     ). Write "no solution" as your answer.

8. 8

   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.

   • Example: We found two solutions, and . One of our lines has the equation . Plug in
     
     and
     
     , then solve each equation to find that
     
     and
     
     .

9. 9

   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.

   • Example: When we plugged in , we got , so one intersection is at (2, 9). The same process for our second solution tells us another intersection lies at (-3, 4).

• 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.

Co-authored by:

Assistant Professor of Mathematics

This article was co-authored by Mario Banuelos, PhD. 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. This article has been viewed 1,029,927 times.

Co-authors: 21

Updated: February 8, 2022

Views: 1,029,927

Article Rating: 69% - 27 votes

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