Solving Linear Equations Lesson

Solving Linear Equations simply means finding the value of the variable(s) in any given linear equation(s) or system of equations. A linear equation is an equation with a degree of one. We will first review solving linear equations with one variable, and then discuss solving linear equations with two variables, also known as systems of equations.

Solving Linear Equations in one variable means we will get one solution for that variable. To solve a linear equation we isolate the variable by moving the variable term to one side of the equation and the constant to the other side of the equation. For example, if we have 2x + 4 = 6, we will start by subtracting the 4 from both sides to keep the equation balanced, which gives us 2x = 2. Then divide both sides by 2 to get the variable x, by itself. This results in x = 1.

Solving Linear Equations with two variables requires two equations, called a system of equations. There are three methods for solving systems of equations, substitution, elimination, and graphing.

Substitution Method – Involves solving one of the equations for one variable and substituting that expression into the other equations, gradually reducing the system to a single equation in one variable.

Elimination Method – Involves adding or subtracting multiples of the equations to eliminate one of the variables, resulting in a simpler system that can be solved more easily.

Graphical Method – Involves graphing each equation on the same coordinate plane and finding the point(s) of intersection, which represent the solution(s) to the system.

The substitution method requires isolating one variable in one equation, and then substituting the equivalent expression into the second equation. Once we substitute that equivalent expression into the second equation, we have an equation with only one variable, in which we can find the value of that variable. Then we substitute that value into one of the original equations to find the value of the other variable. Let’s look at an example to better understand this process.

Given the following system of linear equations, we will use substitution to solve for the two variables.

Notice that the variable, x, in the second equation does not have a coefficient, therefore, we can easily get that variable by itself, by adding the 11 to both sides of the equation. We will then substitute the expression 2y + 11, into the x into the first equation, and solve that equation for y, as shown below.

Step 1 – Isolating a Variable

Step 2 – Substitute the value into the other equation and solve for the one variable. Next, we will distribute the 7 into the parentheses, combine like terms and solve for y.

Step 3 – Substitute the value of ‘y’ into one of the original equations to solve for x.

Therefore, the solution to this system of linear equations is (3, -4).

Solving Linear Equations using elimination is convenient when both equations are set up in standard form, Ax + By=C, and we can easily add or subtract the two equations to eliminate a variable. We can also use multiplication in order to eliminate a variable. Let’s solve the following system of equations using elimination.

To begin, check to make sure that both equations are in standard form, Ax + By = C. Then determine if we can simply add or subtract the two equations to eliminate either the x or the y variable. In this case we cannot, therefore, we will have to multiply by a constant in order to eliminate one of the variables. Notice that the y term in the second equation does not have a coefficient, which makes it easier to use multiplication and eliminate the y-variable. We will multiply the bottom equation by −3, that way we can add the two equations, and eliminate the y-variable.

Now we will add the new equation to the original equation, as shown below.

Now we will substitute the x value in to one of the original equations, and solve for y.

Therefore, by solving the system of equations, we get the solution x = -3 and y = -2. Written as an ordered pair, (-3, -2).

The final method for solving linear equations is by graphing. To solve by graphing both equations should be in slope-intercept form, y= mx + b. Then you graph both equations, and find the intersection point. The point of intersection is the solution to the system of linear equations.

Let’s look at the equations:

In order to graph these equations, we need to get the second equation into slope intercept form, by subtracting the 2x from both sides. Giving us the system of equations:

Now that both equations are in slope intercept form, we can graph them using their y-intercept and slope.

From the graph above, we can see that the two lines intersect at the point (3, 2). Therefore the solution to the system of equations is x = 3 and y = 2.

To review there are three methods of solving systems of linear equations. Each method has advantages and disadvantages, and can be easier to use depending on how the equations are set up. When both equations are in slope intercept form, graphing or substitution are the best options. If both equations are in standard form, then it is recommended that elimination is used. Lastly, if one equation has a variable isolated, then substitution is the most ideal method for solving the equations.

Examples

Example 1 – Solve the following system of equations:

Solution: In order to solve this system of equations, I would use the graphical method since both equations are in slope intercept form. For the red line, top equation, I would begin by plotting a point at the y-intercept of -4, then from that point, I would do the slope of 5/4. To do the slope, from the y-intercept, you would go up 5 units, and right 4 units. For the blue line, bottom equation, I would begin by plotting a point at the y-intercept which is 2, then do the slope of -1/4, by going down 1 unit and right 4 units.

By graphing both equations, we will identify the point of intersection.

We notice that the point of intersection of the two equations is (4, 1), which means that the solution is x = 4 and y = 1.

Example 2 – Solve the system of equations:

Solution: Start by recognizing that in the top equation the y-variable is isolated, this indicates that the best method to use to solve this system of linear equations is substitution. Therefore, we will take the expression -2x + 3 and substitute it in for the y in the second equation. Then we will solve that second equation for x.

Now that we have solved for x, we will take that value and substitute it back into either of the equations to solve for y.

In solving this system of linear equations, we get that the solution is x = 2 and y = −1.

Example 3 – Solve the system of equations:

Solution: Start by recognizing that both equations are in standard form, which means we will likely need to use the elimination method. By analyzing the constants with each variable, I notice that I cannot simply add or subtract the two equations to eliminate a variable, and therefore I will have to multiply each equation by a constant. Since the x terms are already opposite signs, the top is positive and the bottom is negative, I am choosing to eliminate the x variable by multiplying each term in the top equation by 5, and each term in the bottom equation by 4. Then I will add the two equations together to eliminate the x variable and solve for y.

Now we will take this y-value and substitute it back into one of the original equations to solve for x.

We can see by solving this system of equations that the solution is x = 2 and y = -7, or written as an ordered pair (2, -7).

FAQs on Solving Linear Equations

1) What is a linear equation with two variables?

A linear equation with two variables is an equation in the form Ax+By=C, where A, B, and C are constants, and x and y are the variables.

2) How do I solve a linear equation with two variables?

To solve a linear equation with two variables, you typically need at least two equations. You can use various methods like substitution, elimination, or graphing to find the values of the variables that satisfy both equations simultaneously.

3) What is the substitution method?

The substitution method involves solving one of the equations for one variable and substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can then be solved to find the value of that variable. Once you have the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable.

4) What is the elimination method?

Th eelimination method involves adding or subtracting multiples of the equations to eliminate one of the variables, resulting in a simpler system that can be solved more easily. You aim to add or subtract the equations in a way that cancels out one of the variables when you add or subtract the equations.

5) What is the graphical method?

The graphical method involves graphing each equation on the same coordinate plane and finding the point(s) of intersection, which represent the solution(s) to the system. This method is visually intuitive but may not always be accurate, especially for systems with fractional or irrational solutions.

6) How do I know if a system of linear equations has a solution?

A system of linear equations can have one solution, no solution, or infinitely many solutions. You can determine the number of solutions by analyzing the slopes and intercepts of the equations. If the lines represented by the equations intersect at a single point, there is one solution. If the lines are parallel and do not intersect, there is no solution. If the lines coincide, there are infinitely many solutions.

7) What if the system is inconsistent?

If a system of linear equations is inconsistent, it means there is no solution that satisfies all the equations simultaneously. This can happen if the equations represent parallel lines or if they contradict each other.