An algebraic expression that has a maximum degree of 1 is known as a linear equation. Lines in the coordinate system are defined by these equations. Linear equations that have only one variable or a homogeneous variable then it is known as a linear equation in one variable. To make a linear equation true, we have to substitute the values for the unknown variables. When we have the case of an equation with one variable, there is only one solution. In other words, the linear equation represents a straight line. But if we have two-variable linear equations, the Cartesian coordinates of a point of the Euclidean plane are used to calculate the solution.
Linear Equation in One Variable
Linear equations in one variable can be expressed in the standard form of ax + b = 0. a and b represent two integers, and x is a variable. Such an equation yields only one solution. Example, 2y + 7 = 3, ½ x – 4 = 2x etc.
Solving Linear Equations in One Variable
The steps required to solve linear equations in one variable are listed as follows
- Group all the variable terms on one side of the equation while the other constant terms to the other side.
- If there are fractional terms, multiply the equation with the denominators.
- Add or subtract the like terms to reduce the equation to a form with one variable and one constant term.
- Divide the constant term with the coefficient number to give the solution of the equation.
Example: Find the value of x in the equation 7/2x – 4 = 10.
Step 1: Shift – 4 to the right-hand side of the equation; 7/2x = 10 + 4 = 14.
Step 2: Multiply the entire equation with the denominator 2; 7x = 28.
Step 3: Divide the equation with the coefficient 7; x = 3.
Thus, x = 3 is the solution to this linear equation.
Linear Equations in Two Variables
Linear equations in two variables are used to explain the geometry of lines or the graph of two lines. These can give the solution to the given equations. The standard form of linear equation in two variables is given by ax + by = c. x, y are variables, a and b are coefficients, and c is the constant term.
Steps to Solve Linear Equations in 2 Variables
- Write down both equations one after another with the x and y term on one side and the constant term on the other side of the equation.
- To eliminate the x term, we have to multiply one equation with the x coefficient of the other equation and vice versa.
- We then subtract or add both equations to remove the x terms.
- Solve for the y term.
- Repeat the above steps to eliminate the y terms to get an answer to x.
Example: Find x and y in 2x + 3y = 5; x – y = 10.
Step 1: As the equation is already written in the standard form, we multiply the second equation by 2; 2x – 2y = 20.
Step 2: Now we subtract this equation from the first; 3y + 2y = – 15
Step 3: y = – 3.
Step 4: Multiply the second equation with 3 and add it to the first equation to eliminate y. Thus, we get x = 3.
Hence, the solution to our equations is x = 3 and y = -3.
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