Imagine having a budget of £1000 and craving a £25 slice of pizza and £20 donuts. To determine how many of each item can be purchased, we must turn to math and represent the scenario with the equation 25x + 20y = 1000, where x is the number of pizza slices and y is the number of donuts. In this comprehensive guide, we will delve into the world of linear equations, exploring various techniques to solve them and confirm their solutions.

Linear equations, also known as one-degree equations, are algebraic expressions where the highest power of the variable is always 1. They can be written in standard form as ax + b = 0 for one variable, and ax + by = c for two variables. In both cases, x and y have a power of 1, resulting in a straight line when graphed.

The ultimate goal when solving linear equations is to find the values of the variables that make the equation true when substituted back into the original expression. The fundamental rule to solve linear equations is known as "the golden rule," which states to do the same to both sides of the equation to maintain its balance.

Linear equations in one variable can be solved by grouping like terms and isolating the variable through operations on both sides. The following steps are involved:

- Simplify each side of the equation (if necessary).
- Isolate the variable by grouping like terms.
- Calculate the value of the variable using algebraic methods.
- Verify the solution by substituting it back into the original equation.

For example, let us solve the equation 3x + 2 = 0.

**Solution:**

3x + 2 = 0

Subtract 2 from both sides of the equation to isolate the variable:

3x + 2 - 2 = 0 - 2

Simplify to get:

3x = -2

Divide both sides of the equation by 3 to obtain the value of x:

3x / 3 = -2 / 3

Therefore, x = -2/3

To verify the solution, substitute the value of x back into the original equation:

3(-2/3) + 2 = 0

-2 + 2 = 0

0 = 0

**Conclusion:** The solution, x = -2/3, satisfies the original equation.

Next, let us solve the equation x + 7 = 18.

**Solution:**

x + 7 = 18

Subtract 7 from both sides to isolate the variable:

x + 7 - 7 = 18 - 7

Simplify to get:

x = 11

Therefore, x = 11

To verify the solution, substitute the value of x back into the original equation:

11 + 7 = 18

18 = 18

**Conclusion:** The solution, x = 11, satisfies the original equation.

Solving linear equations in two variables requires an additional equation with the same variables to obtain absolute values. One technique for solving this type of equation is substitution. It involves making one variable the subject of one of the equations and substituting its value into the other equation to obtain a single variable to solve for.

For instance, let us solve for x and y given the equations 2x + 5y = 20 and 3x + 5y = 12.

**Solution:**

2x + 5y = 20

3x + 5y = 12

Make y the subject of the first equation by subtracting 2x:

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

Substitute this value for y into the second equation:

3x + 5(20 - 2x) = 12

Solve for x:

3x + 100 - 10x = 12

-7x + 100 = 12

-7x = -88

x = 88/7 = 12.57

To find the value of y, substitute the value of x into any of the original equations:

2(12.57) + 5y = 20

25.14 + 5y = 20

5y = -5.14

y = -1.03

**Conclusion:** The solutions, x = 12.57 and y = -1.03, satisfy both original equations.

When working with linear equations in two variables, we are searching for a solution that satisfies both equations.

Linear equations may seem daunting, but they are actually the simplest type of equations to solve. One method is by graphing the equations on the same coordinate plane to find their point of intersection. Let's take a closer look at how to do this using an example.

In this example, we will solve the equation **y - 2x = 2-x = -y-1** by graphing it.

**The Solution:**

The first step is to rewrite both equations in the slope-intercept form, **y = mx + b**, which helps us determine the slope and y-intercept of each line.

- Equation 1:
**y - 2x = 2**becomes**y = 2x + 2** - Equation 2:
**-x = -y-1**becomes**y = x - 1**

Next, we need to find two points on each line by assigning values to x. For this example, let's use x = 1 and x = 2. By plugging these values into each equation, we can determine the corresponding y-values.

- For Equation 1: when x = 1, y = 4; when x = 2, y = 6.
- For Equation 2: when x = 1, y = 0; when x = 2, y = 1.

Now that we have the necessary points, we can plot both lines on the same coordinate plane and identify their point of intersection. In this example, the solution is (–3, –4).

To confirm that this is the correct solution, we can substitute these values back into the original equations and see if they hold true.

- For Equation 1:
**y-2x = 2**becomes**-4-2(-3) = 2**, which simplifies to**2 = 2**. - For Equation 2:
**-x = -y-1**becomes**-(-3) = -(-4)-1**, which simplifies to**3 = 3**.

Therefore, our solution is correct.

**Key Takeaways:**

- Linear equations are equations where the highest power of the variable is 1.
- In one variable, they are in the form
**ax + b = 0**. In two variables, they are in the form**ax + by = c**. - Solving linear equations in one variable involves isolating the variable and performing necessary arithmetic.
- Solving linear equations in two variables requires finding a definitive solution for both variables.

In simple terms, a linear equation is an equation that forms a straight line when graphed. This means that its highest variable power is 1.

**2x - 4 = 75** and **5 - 4y - 3 = 12**

There are three main methods for solving linear equations: graphing, substitution, and elimination.

To effectively solve linear equations, follow these simple steps:

- Combine like terms and simplify both sides of the equation.
- Isolate the variable term by using addition or subtraction.
- Solve for the variable by using multiplication or division.

If you're struggling with solving equations, you're not alone! Many people find them intimidating and confusing. However, linear equations in one variable are actually the easiest type to solve, regardless of their form.

To begin, identify the variable in the equation, which is represented by letters like x, y, or z. Next, isolate the variable on one side of the equation. This can be achieved by using the inverse operation or by adding/subtracting the variable terms from both sides to keep the equation balanced.

Once the variable is isolated, solve for it by performing any necessary arithmetic operations. And just like that, you have solved a linear equation in its simplest form!

In conclusion, don't let equations intimidate you. Linear equations in one variable are the easiest to solve, no matter their form. With these steps in mind, you can tackle any linear equation with confidence and ease.

Solving linear equations can seem daunting at first, but with a few simple tips, you'll be able to tackle them with ease. The key is to identify the variable and use inverse operations or addition/subtraction to isolate it. With practice and perseverance, you'll become a pro at solving linear equations in no time.

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