How Do You Know How Many Solutions a System of Equations Has Linear Algebra

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Solving a organization of equations requires you lot to discover the value of more ane variable in more than one equation. You can solve a system of equations^[i] through addition, subtraction, multiplication, or commutation. If yous desire to know how to solve a system of equations, just follow these steps.

1. ane

   Write 1 equation above the other. Solving a organization of equations by subtraction is ideal when you encounter that both equations have 1 variable with the same coefficient with the same charge.^[2]

   • For instance, if both equations take the variable positive 2x, y'all should employ the subtraction method to find the value of both variables.
   • Write 1 equation to a higher place the other by matching up the x and y variables and the whole numbers. Write the subtraction sign outside the quantity of the second arrangement of equations.
   • Ex: If your two equations are 2x + 4y = 8 and 2x + 2y = ii, and then you should write the first equation over the second, with the subtraction sign outside the quantity of the second organization, showing that you'll be subtracting each of the terms in that equation.
     • 2x + 4y = 8
     • -(2x + 2y = 2)

2. two

   Subtract like terms. Now that you've lined upwardly the ii equations, all you have to do is subtract the like terms. You lot tin take it one term at a fourth dimension:

   • 2x - 2x = 0
   • 4y - 2y = 2y
   • 8 - 2 = 6
     • 2x + 4y = viii -(2x + 2y = two) = 0 + 2y = vi

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3. 3

   Solve for the remaining term. In one case you've eliminated one of the variables by getting a term of 0 when y'all subtract variables with the aforementioned coefficient, you should just solve for the remaining variable by solving a regular equation. You tin remove the 0 from the equation since information technology won't change its value.

   • 2y = six
   • Divide 2y and 6 past ii to get y = three

4. four

   Plug the term back into one of the equations to notice the value of the first term. Now that you know that y = 3, you just accept to plug it in to one of the original equations to solve for ten. It doesn't matter which one you lot cull because the answer will be the same. If 1 of the equations looks more complicated than the other, merely plug information technology into the easier equation.

   • Plug y = 3 into the equation 2x + 2y = ii and solve for x.
   • 2x + 2(three) = ii
   • 2x + 6 = 2
   • 2x = -4
   • x = - two
     • You have solved the system of equations by subtraction. (x, y) = (-2, 3)

5. 5

   Cheque your answer. To brand sure that you solved the organization of equations correctly, you can merely plug in your 2 answers to both equations to make sure that they work both times. Here's how to practice it:

   • Plug (-two, 3) in for (x, y) in the equation 2x + 4y = 8.
     • 2(-ii) + 4(3) = viii
     • -iv + 12 = 8
     • 8 = 8
   • Plug (-two, iii) in for (10, y) in the equation 2x + 2y = ii.
     • 2(-two) + two(3) = two
     • -4 + 6 = 2
     • ii = ii

   Advert

1. 1

   Write one equation above the other. Solving a system of equations by add-on is platonic when you see that both equations have one variable with the same coefficient with opposite charges. For instance, if one equation has the variable 3x and the other has the variable -3x, and then the improver method is ideal.^[iii]

   • Write one equation above the other past matching upward the 10 and y variables and the whole numbers. Write the addition sign outside the quantity of the second system of equations.
   • Ex: If your 2 equations are 3x + 6y = 8 and x - 6y = 4, so you should write the outset equation over the second, with the addition sign outside the quantity of the 2nd system, showing that you'll be adding each of the terms in that equation.
     • 3x + 6y = 8
     • +(x - 6y = iv)

2. 2

   Add together similar terms. Now that you've lined up the ii equations, all you have to do is add the similar terms. You can accept information technology one term at a time:

   • 3x + 10 = 4x
   • 6y + -6y = 0
   • 8 + 4 = 12
   • When you lot combine information technology all together, you get your new product:
     • 3x + 6y = viii
     • +(x - 6y = 4)
     • = 4x + 0 = 12

3. iii

   Solve for the remaining term. Once you've eliminated one of the variables by getting a term of 0 when you subtract variables with the aforementioned coefficient, you should merely solve for the remaining variable past solving a regular equation. You lot tin remove the 0 from the equation since information technology won't modify its value.

   • 4x + 0 = 12
   • 4x = 12
   • Split 4x and 12 by 3 to get x = 3

4. four

   Plug the term dorsum into the equation to find the value of the kickoff term. Now that y'all know that x = iii, you merely have to plug it into one of the original equations to solve for y. It doesn't matter which i you choose because the respond volition exist the same. If one of the equations looks more complicated than the other, just plug it into the easier equation.

   • Plug x = iii into the equation 10 - 6y = 4 to solve for y.
   • iii - 6y = 4
   • -6y = 1
   • Dissever -6y and ane by -6 to get y = -i/6
     • You have solved the arrangement of equations by addition. (10, y) = (3, -ane/6)

5. 5

   Check your answer. To make sure that you solved the system of equations correctly, you lot can just plug in your 2 answers to both equations to make sure that they work both times. Here's how to exercise information technology:

   • Plug (iii, -1/6) in for (10, y) in the equation 3x + 6y = 8.
     • 3(3) + 6(-one/6) = 8
     • 9 - 1 = viii
     • 8 = 8
   • Plug (3, -1/6) in for (x, y) in the equation x - 6y = 4.
     • iii - (half-dozen * -1/vi) =4
     • 3 - - 1 = iv
     • 3 + 1 = 4
     • 4 = 4

   Advert

1. i

   Write one equation above the other. Write ane equation above the other by matching up the 10 and y variables and the whole numbers. When you employ the multiplication method, none of the variables volition take matching coefficients -- even so.^[4]

   • 3x + 2y = 10
   • 2x - y = ii

2. 2

   Multiply one or both equations until one of the variables of both terms have equal coefficients. Now, multiply one or both of the equations past a number that would make one of the variables accept the aforementioned coefficient. In this instance, you tin multiply the entire 2nd equation by 2 so that the variable -y becomes -2y and is equal to the first y coefficient. Here's how to do it:

   • 2 (2x - y = ii)
   • 4x - 2y = 4

3. iii

   Add or subtract the equations. At present, just use the addition or subtraction method on the 2 equations based on which method would eliminate the variable with the same coefficient. Since you're working with 2y and -2y, you should employ the addition method because 2y + -2y is equal to 0. If you were working with 2y and positive 2y, then you lot would utilize the subtraction method. Hither'due south how to use the improver method to eliminate 1 of the variables:

   • 3x + 2y = 10
   • + 4x - 2y = 4
   • 7x + 0 = 14
   • 7x = 14

4. four

   Solve for the remaining term. Merely solve to find the value of the term that you haven't eliminated. If 7x = 14, then ten = 2.

5. 5

   Plug the term back into the equation to find the value of the first term. Plug the term back into one of the original equations to solve for the other term. Option the easier equation to do it faster.

   • ten = 2 ---> 2x - y = 2
   • four - y = 2
   • -y = -ii
   • y = two
   • You have solved the organization of equations by multiplication. (x, y) = (2, two)

6. 6

   Check your respond. To cheque your respond, just plug the two values you constitute back into the original equations to make sure that y'all have the right values.

   • Plug (two, two) in for (ten, y) in the equation 3x + 2y = 10.
   • 3(2) + 2(2) = 10
   • half-dozen + iv = 10
   • 10 = 10
   • Plug (2, 2) in for (ten, y) in the equation 2x - y = two.
   • two(2) - 2 = 2
   • 4 - 2 = ii
   • 2 = ii

   Advertizement

1. 1

   Isolate one variable. The commutation method is ideal when one of the coefficients in ane of the equations is equal to one. Then, all you have to do is isolate the unmarried-coefficient variable on ane side of the equation to find its value.^[5]

   • If y'all're working with the equations 2x + 3y = 9 and x + 4y = two, you should isolate ten in the second equation.
   • x + 4y = 2
   • ten = two - 4y

2. 2

   Plug the value of the variable you isolated dorsum into the other equation. Take the value you found when you isolated the variable and replace that value instead of the variable in the equation that y'all did not manipulate. You won't exist able to solve annihilation if you plug it back into the equation you just manipulated. Hither'south what to practise:

   • ten = ii - 4y --> 2x + 3y = 9
   • ii(2 - 4y) + 3y = 9
   • 4 - 8y + 3y = nine
   • four - 5y = 9
   • -5y = 9 - four
   • -5y = 5
   • -y = one
   • y = - ane

3. 3

   Solve for the remaining variable. Now that you know that y = - 1, just plug that value into the simpler equation to find the value of x. Here's how you practice information technology:

   • y = -1 --> 10 = 2 - 4y
   • x = two - iv(-1)
   • x = 2 - -4
   • 10 = 2 + iv
   • x = 6
   • Yous accept solved the system of equations by substitution. (x, y) = (6, -i)

4. 4

   Bank check your piece of work. To make sure that you solved the system of equations correctly, you can just plug in your ii answers to both equations to brand sure that they work both times. Here's how to do it:

   • Plug (six, -1) in for (x, y) in the equation 2x + 3y = nine.
     • 2(6) + 3(-one) = 9
     • 12 - 3 = 9
     • 9 = ix
   • Plug (half-dozen, -one) in for (x, y) in the equation x + 4y = 2.
   • half dozen + iv(-i) = 2
   • six - 4 = ii
   • 2 = 2

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Add together New Question

• Question

  What is the value of two numbers if their sum is 14 and their difference is 2?

  

  ten + y = 14. x - y = 2. Add the ii equations together: 2x = 16. ten =eight. 8 - y = 2. y = 6.

• Question

  How do I depict the straight line of y = x2 - 7x + 12 and find zeroes of information technology?

  

  y = ten² - 7x + 12 = (x - three)(x - four). This is a parabola, not a straight line. To detect the zeroes, set (x - 3)(ten - iv) equal to zero. That ways either (10 - 3) or (x - 4) must equal zero. If (ten - iii) equals cypher, x has to equal three. If (x - 4) equals goose egg, x has to equal four. So the zeroes are three and 4.

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• Y'all should me able to solve whatsoever linear system of equations using the addition, subtraction, multiplication, or exchange method, but one method is normally the easiest depending on the equations.^[6]

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Article Summary 10

To solve a system of equations past elimination, make sure both equations have i variable with the same coefficient. Subtract the like terms of the equations then that you're eliminating that variable, then solve for the remaining one. Plug the solution back into one of the original equations to solve for the other variable. To solve by substitution, solve for 1 variable in the start equation, then plug the value into the 2d equation and solve for the second variable. Finally, solve for the first variable in either of the starting time equations. Write your reply by placing both terms in parentheses with a comma between. If you want to learn how to check your answers, keep reading the commodity!

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