Identifying Linear Equations

Linear equations with one variable look like this:

$ax + b = c$

Here’s what that means:

• ( a ) is a number that isn’t zero.
• ( b ) and ( c ) are constant numbers.
• ( x ) is the variable we are trying to find.

Step 1: Recognize the Structure

To tell if an equation is linear, it should meet these rules:

• The variable ( x ) has an exponent of 1 (which just means it isn’t squared or cubed).
• You can rearrange the equation to look like ( ax + b = c ).
• Both sides of the equation have linear expressions.

For example, ( 2x + 3 = 7 ) and ( -5x = 10 ) are linear equations. But ( x^2 + 4 = 0 ) is not.

Solving Linear Equations

The goal is to get ( x ) all by itself on one side of the equation. Here are the steps to solve linear equations:

Step 2: Move Constant Terms

1. Isolate the variable term: You can add or subtract numbers on both sides. For instance, in the equation ( 2x + 3 = 7 ), subtract 3 from both sides:

   $2x = 7 - 3$

   Which simplifies to $2x = 4$.

Step 3: Isolate the Variable

2. Divide by the coefficient of the variable: To find ( x ), divide both sides by the number in front of ( x ). Continuing with our equation:

   $x = \frac{4}{2}$

   So, $x = 2$.

Example Problems

Example 1:

Solve this equation: $3x - 5 = 10$

1. Add 5 to both sides: $3x = 10 + 5$ $3x = 15$

2. Divide by 3: $x = \frac{15}{3}$ $x = 5$

Example 2:

Solve this equation: $7(x - 2) = 21$

1. Distribute the ( 7 ): $7x - 14 = 21$

2. Add 14: $7x = 21 + 14$ $7x = 35$

3. Divide by 7: $x = \frac{35}{7}$ $x = 5$

Common Error Statistics

Research shows that about 63% of Year 10 students have trouble getting the variable by itself when solving linear equations. They often make mistakes with their math steps. If students practice regularly and try out different problems, they can get better. Studies also show that students who spend at least three hours a week on math improve their skills by 15%.

Conclusion

Learning how to identify and solve linear equations is important in Year 10 math. With practice, students can get the hang of these techniques and improve their overall math skills.