System of Equations Solver Calculator

Published on: December 1, 2024
Final Answer: Free • Full Steps: Plus

This System of Equations Solver Calculator finds the one pair of values that satisfies two linear equations at once, and shows the working line by line. Both equations are put in standard form so their coefficients line up, and one variable is then scaled until it matches in both. Subtracting removes that variable, which leaves a single equation in one unknown. Following the same pattern every time keeps the elimination easy to follow.

Step-by-step method

  1. Set up the problem.
  2. Rewrite both equations in standard form.
  3. Choose the variable to eliminate.
  4. Multiply each equation so that variable's coefficients match.
  5. Subtract one equation from the other to eliminate it.
  6. Solve for the variable that is left.
  7. Substitute back to find the other variable.

Formula:

\(a_{1}x + b_{1}y = c_{1},\quad a_{2}x + b_{2}y = c_{2}\)
See Example 1 Hide Example 1

Problem

\(3x + 2y = 16,\;\; 5x - 2y = 8\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Rewrite both equations in standard form.
  3. Choose the variable to eliminate.
  4. Multiply each equation so that variable's coefficients match.
  5. Subtract one equation from the other to eliminate it.
  6. Solve for the variable that is left.
  7. Substitute back to find the other variable.

Step 1

Step 1 - Set up the problem.

In this problem: The system is \(3x + 2y = 16\) and \(5x - 2y = 8\), and both \(x\) and \(y\) are unknown.

\(3x + 2y = 16\)
\(5x - 2y = 8\)

Step 2

Step 2 - Rewrite both equations in standard form.

In this problem: Moving every term to the left of each equals sign gives \(3 x + 2 y - 16 = 0\) and \(5 x - 2 y - 8 = 0\), which lines the coefficients up.

\(3 x + 2 y - 16 = 0\)
\(5 x - 2 y - 8 = 0\)

Step 3

Step 3 - Choose the variable to eliminate.

In this problem: The coefficients of \(x\) are \(3\) and \(5\), so \(x\) is the one to remove.

\(x:\; 3,\; 5\)

Step 4

Step 4 - Multiply each equation so that variable's coefficients match.

In this problem: Multiplying the first equation by \(5\) and the second by \(3\) makes the coefficient of \(x\) the same in both.

\(15 x + 10 y - 80 = 0\)
\(15 x - 6 y - 24 = 0\)

Step 5

Step 5 - Subtract one equation from the other to eliminate it.

In this problem: Subtracting the second equation from the first cancels \(x\) and leaves \(16 y - 56 = 0\).

\(16 y - 56 = 0\)

Step 6

Step 6 - Solve for the variable that is left.

In this problem: Only \(y\) is left, and solving gives \(y = \frac{7}{2}\).

\(y = \frac{7}{2}\)

Step 7

Step 7 - Substitute back to find the other variable.

In this problem: Putting \(y = \frac{7}{2}\) back into \(3 x + 2 y - 16 = 0\) gives \(x = 3\).

\(x = 3\)

Final Answer

\(x = 3,\; y = \frac{7}{2}\)
See Example 2 Hide Example 2

Problem

\(2x = 8,\;\; x + y = 10\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Rewrite both equations in standard form.
  3. Choose the variable to eliminate.
  4. Multiply each equation so that variable's coefficients match.
  5. Subtract one equation from the other to eliminate it.
  6. Solve for the variable that is left.
  7. Substitute back to find the other variable.

Step 1

Step 1 - Set up the problem.

In this problem: The system is \(2x = 8\) and \(x + y = 10\), and both \(x\) and \(y\) are unknown.

\(2x = 8\)
\(x + y = 10\)

Step 2

Step 2 - Rewrite both equations in standard form.

In this problem: Moving every term to the left of each equals sign gives \(2 x - 8 = 0\) and \(x + y - 10 = 0\), which lines the coefficients up.

\(2 x - 8 = 0\)
\(x + y - 10 = 0\)

Step 3

Step 3 - Choose the variable to eliminate.

In this problem: The coefficients of \(x\) are \(2\) and \(1\), so \(x\) is the one to remove.

\(x:\; 2,\; 1\)

Step 4

Step 4 - Multiply each equation so that variable's coefficients match.

In this problem: Multiplying the first equation by \(1\) and the second by \(2\) makes the coefficient of \(x\) the same in both.

\(2 x - 8 = 0\)
\(2 x + 2 y - 20 = 0\)

Step 5

Step 5 - Subtract one equation from the other to eliminate it.

In this problem: Subtracting the second equation from the first cancels \(x\) and leaves \(12 - 2 y = 0\).

\(12 - 2 y = 0\)

Step 6

Step 6 - Solve for the variable that is left.

In this problem: Only \(y\) is left, and solving gives \(y = 6\).

\(y = 6\)

Step 7

Step 7 - Substitute back to find the other variable.

In this problem: Putting \(y = 6\) back into \(2 x - 8 = 0\) gives \(x = 4\).

\(x = 4\)

Final Answer

\(x = 4,\; y = 6\)

Example Problems

Click one to load it into the box and solve it.