Linear equations with two variables

Objective:

  • To know the definition and properties of an equation with two variables;
  • To learn how to solve an equation with two variables.

This virtual activity is designed to be used in math lessons in the next chapter:

  • Grade 6. “Linear equation with two variables”.

Theoretical part

A linear equation with two variables x and y (with two unknowns x and y) is an equation of the form a*x+b+y+c=0, where a,b,c are numbers, and a and b are not equal to 0 at the same time. For example, -2x+7y=0; 12x-11y+5=0 are linear equations with two variables. Since in such equations there are two variables, so the solution to equations with two variables is not one but two numbers, which are usually written in parentheses, and the number that we substitute into the equation instead of x, we write in the first place, and y – in the second place: (x, y).

The solution of an equation with two variables a*x+b+y+c=0 is such a pair of numbers, when substituting which instead of x and y in the equation, the correct numerical equality is obtained. For example, the pair of numbers (0,5;1) is a solution to the equation with two variables 12x-11y+5=0.

To solve a linear equation with two variables is to find the set of solutions. 

If a pair of numbers is a solution to an equation with two variables, the pair is also said to satisfy the equation.

Equations with two variables are called equivalent if all the solutions of one equation are equal to the solutions of the other equation.

Virtual experiment

The “Equality Explorer: Two Variables” simulation allows students to explore the conditions that result in equality and inequality when there are two variables present. Students can build a system of equations and develop a meaningful understanding of a system of equations.

Workflow:

Step 1. You are given: 

  • A board on which to display the equation (1);
  • Scales to construct the equation with equating variables and numbers (2);
  • Variables and numbers to construct the equation (3);
  • Eraser to remove the expression standing on the scales (4);
  • Button to adjust the expression above the scales (5);
  • You can set the same numbers in both heads of the scales by pressing the lock button (6);
  • Button to change the x and y values (7);
  • You can save the generated equations in the image panel by pressing the camera button (8);
  • Buttons to refresh, garbage can and display x and y values (9);
  • Reload button (10).

Step 2. Generate an equation in x =1, y=1 values. You can use the button that adjusts the expressions over the weights.

Step 3. Click the camera button and save to the image panel.

Step 4. Change the value of x and make the equation. If necessary, you can use the eraser and button to adjust the expressions over the weights. Save to the image panel.

Step 5. Change the x and y values and make the equation. You can click the lock button and express both sides of the equation with the same numbers. Save in the image panel.

Step 6. Compose the equations by assigning different values to the x and y variables. Fill in the image panel.

Conclusion

Performing the virtual work “Equality Explorer: Two Variables” allowed students to deepen their knowledge of linear equations with two variables and learn how to use them to solve practical problems. The fact that the variable has a different value by balancing it on weights makes it easier for students to visualize and learn how to make different equations.