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MATHCRAVE ALGEBRA
Simultaneous Equations
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AI Simultaneous Equation Solver

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About AI Simultaneous Equations Solver

MathCrave ai simultaneous equations solver is an algebra solver that helps solve systems of simultaneous equations with ease. With this math tool, you can quickly find the values of multiple unknown variables that satisfy all the given equations. It provides step-by-step solutions, guiding you through the entire process and helping you understand the logic behind each calculation.

Math Problems AI Simultaneous Equations Solver Solves

  • Solve simultaneous equations in two unknowns by substitution

  • Solve simultaneous equations in two unknowns by elimination

  • Solve simultaneous equations involving practical situations

  • Solve simultaneous equations in three unknowns

How to Solve Simultaneous Equations in Two Unknowns by Substitution

  • Step 1: Start by solving one of the equations for one variable in terms of the other variable. Choose the equation that appears simpler to isolate a variable.

  • Step 2: Substitute the expression obtained in step 1 into the other equation. This will create an equation with only one variable.

  • Step 3: Solve the equation obtained in step 2 to find the value of the remaining variable.

  • Step 4: Substitute the value found in step 3 back into one of the original equations to solve for the other variable.

  • Step 5: Check the solution by substituting the values of both variables into both original equations. If the values satisfy both equations, then the solution is correct. If not, recheck your work.

 

Worked Example

Solve x + 2y =−1, 4x − 3y = 18 by substitution method

To solve the system of equations using the substitution method, we’ll follow these steps:

Step 1: Solve one of the equations for one variable in terms of the other variable.

  • Let’s solve the first equation, x + 2y = -1, for x:

  • x = -1 – 2y

Step 2: Substitute the expression obtained in step 1 into the other equation and solve for the remaining variable.

  • Substitute x = -1 – 2y into the second equation, 4x – 3y = 18:

  • 4(-1 – 2y) – 3y = 18

  • -4 – 8y – 3y = 18

  • -11y – 4 = 18

  • -11y = 18 + 4

  • -11y = 22

  • y = -2

Step 3: Substitute the value of y into one of the original equations and solve for x.

  • Using the first equation, x + 2y = -1:

  • x + 2(-2) = -1

  • x – 4 = -1

  • x = -1 + 4

  • x = 3

Step 4: Write the solution as an ordered pair (x, y).

  • The solution to the system of equations is (3, -2).

  • Therefore, the solution to the system of equations x + 2y = -1 and 4x – 3y = 18 using the substitution method is x = 3 and y = -2.

 

Solve the System of Equations Using the Elimination Method

Step 1: Multiply one or both of the equations by suitable constants so that the coefficients of one of the variables will cancel out when the equations are added or subtracted.

  • Let’s multiply the first equation, x + 2y = -1, by 4 and the second equation, 4x – 3y = 18, by 1 to eliminate the x terms.

  • The new equations are:

  • 4x + 8y = -4

  • 4x – 3y = 18

Step 2: Subtract the first equation from the second equation to eliminate the x terms.

  • (4x – 3y) – (4x + 8y) = 18 – (-4)

  • -3y – 8y = 18 + 4

  • -11y = 22

  • y = -2

Step 3: Substitute the value of y into one of the original equations and solve for x.

  • Using the first equation, x + 2y = -1:

  • x + 2(-2) = -1

  • x – 4 = -1

  • x = -1 + 4

  • x = 3

Step 4: Write the solution as an ordered pair (x, y).

  • The solution to the system of equations is (3, -2).

  • Therefore, the solution to the system of equations x + 2y = -1 and 4x – 3y = 18 using the elimination method is x = 3 and y = -2.

     

Solving Simultaneous Equation of Three Unknowns

Given equations:

  • Equation 1: x + 2y + 4z = 16

  • Equation 2: 2x – y + 5z = 18

  • Equation 3: 3x + 2y + 2z = 14

Let’s solve Equation 1 for x:

  • x = 16 – 2y – 4z

Now substitute this expression for x in the other two equations. Substituting x in Equation 2:

  • 2(16 – 2y – 4z) – y + 5z = 18

  • 32 – 4y – 8z – y + 5z = 18

  • -5y – 3z = -14 …..(Equation A)

Substituting x in Equation 3:

  • 3(16 – 2y – 4z) + 2y + 2z = 14

  • 48 – 6y – 12z + 2y + 2z =14

  • -4y – 10z = -34 ..…(Equation B)

  • Now we have the following system of equations:

  • Equation A: -5y – 3z = -14

  • Equation B: -4y – 10z = -34

  • We can solve this system using the method of elimination.

  • Multiply Equation A by 4 and Equation B by -5 to eliminate the y coefficients:

  • -20y – 12z = -56

  • 20y + 50z = 170

Add the two equations:

  • (-20y – 12z) + (20y + 50z) = -56 + 170

  • 38z = 114

  • z = 3

Substitute the value of z in Equation A to solve for y:

  • -5y – 9 = -14

  • -5y = -5

  • y = 1

Substitute the values of y and z in Equation 1 to solve for x:

  • x + 2(1) + 4(3) = 16

  • x + 2 + 12 = 16

  • x + 14 = 16

  • x = 2

Therefore, the solution to the system of equations is:

  • x = 2

  • y = 1

  • z = 3