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# Solving Linear Equations

Assuming we were to order a slice of pizza that cost £25, and donuts that also costs £20 for one. If all the money we budgeted was £1000, how much of each can we order?

This system can be modelled mathematically into a linear equation as

$25x+20y=1000$,

where x and y could be found, considering$x=numberofpizzaslices$ and $y=numberofdonuts$.

In this article, we will learn about solving linear equations, using different methods to solve them, and how to verify their solutions.

## What is a linear equation?

A linear equation, also known as a one-degree equation, is an equation in which the highest power of the variable is always 1.

Linear equations in one variable are in standard form as

$ax+b=0$,

where x is a variable, a is a coefficient and b is a constant.

They are in two variable standard form as

$ax+by=c$,

where x and y are variables, c is a constant, and a and b are coefficients.

They are linear because both their variables have power 1, and graphing these equations is always in a straight line.

Solving linear equations involves finding the values of the variables such that the equation is satisfied when they are substituted back into them. The fundamental rule that is required to solve them is "the golden rule". This states that you do unto one side of the equation what you do unto the other side of the equation.

## Linear equations in one variable

Linear equations in one variable as discussed earlier in this article are in the form

$ax+b=0$,

where x is a variable, a is a coefficient and b is a constant.

These equations are solved easily by grouping like terms first. This means the terms with the variable will be sent to one side of the equation, whilst the constants go to the other side. Then, they can now be operated to find the value of the variable.

The steps that are associated with solving linear equations in one variable are:

1. Simplify each side of the equation if need be;

2. Isolate the variable;

3. Algebraically find the value of the variable;

4. Verify your solution by substituting the value back into the equation.

Let us take an example below.

Solve the equation$3x+2=0$.

Solution

$3x+2=0$

Each side of the equation is simplified, step 1 is achieved.

Step 2: Group like terms by subtracting 2 from each side of the equation

$3x+2-2=0-2$

$3x=-2$

Step 3: Divide each side by 3

$\frac{3x}{3}=\frac{-2}{3}$

$x=\frac{-2}{3}$

Step 4: Now we can evaluate this to see if this is true. The equation means that everything on the left side should be equal to what is on the right. Hence, everything on the left side of the equation should be equal to 0. We will substitute the solution into the equation now.

$3x+2=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}3\left(\frac{-2}{3}\right)+2=0$

We will now divide 3 outside the bracket by the 3 as the denominator, and we will have 1 each.

$-2+2=0$

$0=0$

We see here that the solution we have is true.

Solve the equation $x+7=18$.

Solution

Each side of the equation is simplified, step 1 is achieved.

Step 2 and 3: Group like terms by subtracting 7 from each side of the equation

$x+7-7=18-7$

$x=11$

Step 4: Now we can evaluate this to see if this is true. The equation means that everything on the left side should be equal to what is on the right. Hence if we add x to 7, we should have 18

$x+7=18$

$x=11$

$11+7=18$

This means our equation is true.

## Linear equations in two variables

Solving linear equations in two variables can no longer give you absolute values unless another equation is provided that possesses the same variables as the first equation. For instance, if we were given an equation as

$x+y=5$,

then, if x = 3, y = 2, if x = 4, y = 1, and so on.

The only way we can have absolute values is to have another equation with the same variables.

One way to solve this type of equation is by substitution method. You make one variable the subject of one of the equations and substitute that value into the other equation to have only one variable to find. We can take the example below.

Solve for x and y given the equations $2x+5y=20$ and $3x+5y=12$.

Solution

$\left\{\begin{array}{l}2x+5y=20\\ 3x+5y=12\end{array}\right\$

Let us make y the subject of the first equation by subtracting 2x from each side of the equation.

$2x-2x+5y=20-2x$

$5y=20-2x$$\frac{5y}{5}=\frac{20}{5}-\frac{2x}{5}$$y=4-\frac{2x}{5}$

Now we will substitute this value of y into the second equation

$3x+5y=12$

$3x+5\left(4-\frac{2x}{5}\right)=12$$3x+20-2x=12$$x=12-20$$x=-8$

We will now substitute this value for x into any of the equations to find y. We will use the first.

$2x+5y=20$

$2\left(-8\right)+5y=20$$-16+5y=20$

Add 16 to each side of the equation to make 5y stand alone on that side of the equation

$16-16+5y=20+16$

$5y=36$

Divide through by 5 to find y

$\frac{5y}{5}=\frac{36}{5}\phantom{\rule{0ex}{0ex}}$

$y=\frac{36}{5}$

### Solving linear equations in two variables by graphing

Linear equations in two variables are such that both equations would remain true when we find a solution for each variable. When we want to solve systems of linear equations by graphing, we plot both equations on the same coordinate plane. Now the point where both lines intersect is the solution for the system. Let us look at the example below.

Solve the equation

$\left\{\begin{array}{l}y-2x=2\\ -x=-y-1\end{array}\right\$

Solution

As mentioned earlier, we will want to plot both equations on the coordinate plane. We will start by finding the y-intercept and slope for each line. This means for each equation, we will rewrite it in the slope-intercept form. Slope intercept form is given by;

$y=mx+b$

where m is the slope

b is the y-intercept

x is the x-value on the coordinate plane

y is the y-value on the coordinate plane

$y-2x=2$ [Equation 1]

$y=2x+2$

This means that;

$m=2$

$y-intercept=2$

$-x=-y-1$ [Equation 2]

$y=x-1$

This means that;

$m=1$

$y-intercept=-1$

Both equations in the slope-intercept form are given by;

$\left\{\begin{array}{l}y=2x+2\\ y=x-1\end{array}\right\$

Let us find the y-value by assuming two values on the x-axis. Recall that two points are enough to give us a line. Given two values on the x-axis, we will use 1 and 2, what is y when x = 1? And what is y when x = 2?

The solution to these two questions should give us the lines of both equations.

$y=2x+2$.

Substitute 1 into the equation assuming x = 1,

$y=2\left(1\right)+2$

$y=4$

When $x=1$, $y=4$.

Substitute 2 into the equation assuming x = 2,

$y=2\left(2\right)+2$

$y=6$

When $x=2$, $y=6$.

We now have two points for Equation 1 to be plotted.

Graph of line y = 2x + 2 - StudySmarter Originals

The same will be done for the Equation 2,

$y=x-1$.

Substitute 1 into the equation assuming x = 1,

$y=1-1$

$y=0$

When $x=1$, $y=0$.

Substitute 2 into the equation assuming x = 2,

$y=2-1$

$y=1$

When $x=2$, $y=1$.

Let us plot these points and draw the line on the same coordinate plane.

Graph for equations y = 2x + 2 and y = x – 1, StudySmarter Originals

The point they both intercept is the solution for the problem, (–3, –4).

This means

$x=-3$

$y=-4$

Now we can evaluate this to see if this is true. Working with equations means that everything on the left side should be equal to what is on the right. Since we have two equations here, we will verify both. Let us start with the first one.

$y-2x=2$

We will substitute the values we just found into the equation

$-4-2\left(-3\right)=2$

Since both negative values are multiplying each other, the result becomes positive.

$-4+6=2$

$2=2$.

We do see here that the first equation is satisfied. We can go ahead to do the same with the second equation.

$-x=-y-1$

Substitute the values we just found into the equation

$-\left(-3\right)=-\left(-4\right)-1$

Negative values multiplying each other will result in positive.

$3=4-1$

$3=3$

We do realize here that the solution satisfies both equations, therefore, the solution is correct.

## Solving Linear Equations - Key takeaways

• Linear equations are equations that have the highest power of the variable is always 1.
• Linear equations in one variable are in standard form as ax + b = 0, where x is a variable, a is a coefficient and b is a constant.
• They are in two variable standard form as ax + by = c, where x and y are variables, c is a constant, and a and b are coefficients.
• Solving for linear equations in one variable means finding for that variable by making it the subject and performing the necessary arithmetics.
• Solving for linear equations in two variables requires another equation to have an absolute solution for variables.

A linear equation is one in which the highest power of the variable is always 1.

Here are two examples:

2x – 4 = 7

5 – 4y – 3 = 12

Graphing, substitution, and elimination.

The following steps provide a good method to use when solving linear equations.

(1) Simplify each side of the equation by removing parentheses and combining like terms.

(2) Use addition or subtraction to isolate the variable term on one side of the equation.

(3) Use multiplication or division to solve for the variable.

No matter the form of the equation, a linear equation in one variable is the simplest type of equation to be solved.

## Final Solving Linear Equations Quiz

Question

One-dimensional equations are also known as ____

Linear equations

Show question

Question

What are linear equations?

Linear equations are equations that have the highest power of the variable is always 1

Show question

Question

What is the standard form of linear equations in one variable?

ax + b = 0

Show question

Question

What is the standard form for linear equations in two variables?

ax + by = c

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Question

How many equations are needed to solve linear equations in one variables to have absolute values as solutions?

One

Show question

Question

How many equations are needed to solve linear equations in two variables to have absolute values as solutions?

2

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Question

What is the value of x in the equation 2x + 5 = 15?

x = 5

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Question

Find x if 5 - x = 12

x = -7

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Question

What is the value of x in the equation (2x + 5)/(x + 4) = 1

x = -1

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Question

Solve 6x - 19 = 3x - 10

x = 3

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