Algebraic Solution - Examples, Exercises and Solutions

Question Types:
Algebraic Solution: Solving an equation by adding/subtracting from both sidesAlgebraic Solution: Solve using the substitution method

A system of linear equations is a collection of two or more linear equations involving the same variables. The solution to a system of linear equations consists of the values of each of the unknown variables unknowns in the system that satisfy all of its equations, or makes them true.

These questions can be solved in several ways, the algebraic solution consists of two methods:

Substitution method:

  1. Isolate an unknown in any of the equations.
  2. Substitute the unknown that we previously isolated into the other equation of the system in order to determine the value of the unknown.
  3. We insert the value of the unknown that we have discovered in one equation in order to determine the value of the other.

Step-by-step solution of a system of linear equations with two variables: 2X + Y = 5 and 2X + Y = 3. The solution involves elimination to find Y = 2, substitution to solve for X = 1.5, and clear visual formatting for educational clarity. Featured in a guide on solving linear systems algebraically.

Equalization method

  1. We will begin by equating the coefficients in both equations (X X or Y Y )
  2. We then add or subtract one equation from the other and thus eliminate the equal coefficients.
  3. We will proceed to solve the equation with the isolated coefficient for the purpose of determining its value.
  4. Finally we insert the unknown that we have discovered in one equation into the other in order to establish its value.

Step-by-step solution of a system of linear equations with two variables using substitution: 2X - Y = 5 and 2X + Y = 3. The solution involves isolating Y, substituting into the second equation to find X = 2, and then substituting back to find Y = -1. Featured in a guide on solving linear systems algebraically with substitution.

Detailed explanation

Suggested Topics to Practice in Advance

  1. Linear equation with two variables

Practice Algebraic Solution

Question 1

Solve the following system of equations:

\( \begin{cases} x-y=5 \\ 2x-3y=8 \end{cases} \)

Question 2

Solve the above set of equations and choose the correct answer.

\( \begin{cases} -5x+4y=3 \\ 6x-8y=10 \end{cases} \)

Question 3

Solve the above set of equations and choose the correct answer.

\( \begin{cases} -2x+3y=4 \\ x-4y=8 \end{cases} \)

Question 4

Solve the following equations:

\( \begin{cases} 2x+y=9 \\ x=5 \end{cases} \)

Question 5

Solve the following equations:

\( \begin{cases} x+y=18 \\ y=13 \end{cases} \)

Examples with solutions for Algebraic Solution

Exercise #1

Solve the following system of equations:

{x−y=52x−3y=8 \begin{cases} x-y=5 \\ 2x-3y=8 \end{cases}

Video Solution

Answer

x=7,y=2 x=7,y=2

Exercise #2

Solve the above set of equations and choose the correct answer.

{−5x+4y=36x−8y=10 \begin{cases} -5x+4y=3 \\ 6x-8y=10 \end{cases}

Video Solution

Answer

x=−4,y=−414 x=-4,y=-4\frac{1}{4}

Exercise #3

Solve the above set of equations and choose the correct answer.

{−2x+3y=4x−4y=8 \begin{cases} -2x+3y=4 \\ x-4y=8 \end{cases}

Video Solution

Answer

x=−8,y=−4 x=-8,y=-4

Exercise #4

Solve the following equations:

{2x+y=9x=5 \begin{cases} 2x+y=9 \\ x=5 \end{cases}

Video Solution

Answer

x=5,y=−1 x=5,y=-1

Exercise #5

Solve the following equations:

{x+y=18y=13 \begin{cases} x+y=18 \\ y=13 \end{cases}

Video Solution

Answer

x=5,y=13 x=5,y=13

Question 1

Solve the following system of equations:

\( \begin{cases} -8x+5y=3 \\ 10x+y=16 \end{cases} \)

Question 2

Solve the above set of equations and choose the correct answer.

\( \begin{cases} 7x-4y=8 \\ x+5y=12.8 \end{cases} \)

Question 3

Solve the above set of equations and choose the correct answer.

\( \begin{cases} -8x+3y=7 \\ 24x+y=3 \end{cases} \)

Question 4

Find the value of x and and band the substitution method.

\( \begin{cases} -x-2y=4 \\ 3x+y=8 \end{cases} \)

Question 5

Find the value of x and and band the substitution method.

\( \begin{cases} x+y=5 \\ 2x-3y=-15 \end{cases} \)

Exercise #6

Solve the following system of equations:

{−8x+5y=310x+y=16 \begin{cases} -8x+5y=3 \\ 10x+y=16 \end{cases}

Video Solution

Answer

x=1.32,y=2.8 x=1.32,y=2.8

Exercise #7

Solve the above set of equations and choose the correct answer.

{7x−4y=8x+5y=12.8 \begin{cases} 7x-4y=8 \\ x+5y=12.8 \end{cases}

Video Solution

Answer

x=2.33,y=2.09 x=2.33,y=2.09

Exercise #8

Solve the above set of equations and choose the correct answer.

{−8x+3y=724x+y=3 \begin{cases} -8x+3y=7 \\ 24x+y=3 \end{cases}

Video Solution

Answer

x=0.025,y=2.4 x=0.025,y=2.4

Exercise #9

Find the value of x and and band the substitution method.

{−x−2y=43x+y=8 \begin{cases} -x-2y=4 \\ 3x+y=8 \end{cases}

Video Solution

Answer

x=4,y=−4 x=4,y=-4

Exercise #10

Find the value of x and and band the substitution method.

{x+y=52x−3y=−15 \begin{cases} x+y=5 \\ 2x-3y=-15 \end{cases}

Video Solution

Answer

x=0,y=5 x=0,y=5

Question 1

Solve the following system of equations:

\( \begin{cases} 2x-\frac{1}{5}y=18 \\ 3x+y=6 \end{cases} \)

Question 2

Solve the above set of equations and choose the correct answer.

\( \begin{cases} \frac{1}{3}x-4y=5 \\ x+6y=9 \end{cases} \)

Question 3

Find the value of x and and band the substitution method.

\( \begin{cases} -5x+9y=18 \\ x+8y=16 \end{cases} \)

Question 4

Find the value of x and and band the substitution method.

\( (I)-x+3and=12 \)

\( (II)4x+2and=10 \)

Question 5

Solve the above set of equations and choose the correct answer.

\( \begin{cases} \frac{1}{2}x+\frac{7}{2}y=10 \\ -3x+7y=12 \end{cases} \)

Exercise #11

Solve the following system of equations:

{2x−15y=183x+y=6 \begin{cases} 2x-\frac{1}{5}y=18 \\ 3x+y=6 \end{cases}

Video Solution

Answer

x=7.38,y=−16.14 x=7.38,y=-16.14

Exercise #12

Solve the above set of equations and choose the correct answer.

{13x−4y=5x+6y=9 \begin{cases} \frac{1}{3}x-4y=5 \\ x+6y=9 \end{cases}

Video Solution

Answer

x=11,y=−13 x=11,y=-\frac{1}{3}

Exercise #13

Find the value of x and and band the substitution method.

{−5x+9y=18x+8y=16 \begin{cases} -5x+9y=18 \\ x+8y=16 \end{cases}

Video Solution

Answer

x=0,y=2 x=0,y=2

Exercise #14

Find the value of x and and band the substitution method.

(I)−x+3and=12 (I)-x+3and=12

(II)4x+2and=10 (II)4x+2and=10

Video Solution

Answer

x=37,y=297 x=\frac{3}{7},y=\frac{29}{7}

Exercise #15

Solve the above set of equations and choose the correct answer.

{12x+72y=10−3x+7y=12 \begin{cases} \frac{1}{2}x+\frac{7}{2}y=10 \\ -3x+7y=12 \end{cases}

Video Solution

Answer

x=2,y=2.57 x=2,y=2.57

Topics learned in later sections

  1. Two linear equations with two unknowns
  2. Substitution method for two linear equations with two unknowns
  3. Solving with the method of equalization for systems of two linear equations with two unknowns
  4. Solving a System of Linear Equations with Two Variables Using Algebraic Method