Grade 8 math problems: elimination by subtracting equations PDF

📚 Understanding Elimination by Subtraction

Elimination by subtraction is a method used to solve systems of linear equations. A system of linear equations is simply two or more equations with the same variables. The goal of elimination by subtraction is to eliminate one of the variables by subtracting one equation from the other. This leaves you with a single equation with a single variable, which you can then solve.

📜 History and Background

The concept of solving systems of equations has been around for centuries. Ancient civilizations like the Babylonians and Egyptians solved similar problems using various techniques. However, the systematic approach of elimination, including subtraction, became more formalized with the development of algebra.

🔑 Key Principles

• 🎯Identify Matching Coefficients: Look for equations where the coefficient (the number in front of the variable) of either $x$ or $y$ is the same in both equations.
• ➖Subtract the Equations: Subtract one entire equation from the other. Make sure to align the terms correctly (x terms with x terms, y terms with y terms, and constants with constants).
• ✅Solve for the Remaining Variable: After subtracting, you should have an equation with only one variable. Solve for that variable.
• 🔄Substitute Back: Substitute the value you found back into either of the original equations to solve for the other variable.
• ✍️Write the Solution as an Ordered Pair: The solution is usually written as $(x, y)$.

✏️ Real-World Examples

Let's work through a problem to illustrate the process:

Example 1:

Solve the following system of equations:

$2x + y = 7$

$2x - y = 3$

Here, the coefficient of $x$ is the same in both equations.

Subtract the second equation from the first:

$(2x + y) - (2x - y) = 7 - 3$

$2x + y - 2x + y = 4$

$2y = 4$

$y = 2$

Now, substitute $y = 2$ into the first equation:

$2x + 2 = 7$

$2x = 5$

$x = \frac{5}{2}$

So the solution is $(\frac{5}{2}, 2)$.

Example 2:

Solve the following system of equations:

$3x + 2y = 11$

$x + 2y = 5$

Here, the coefficient of $y$ is the same in both equations.

Subtract the second equation from the first:

$(3x + 2y) - (x + 2y) = 11 - 5$

$3x + 2y - x - 2y = 6$

$2x = 6$

$x = 3$

Now, substitute $x = 3$ into the second equation:

$3 + 2y = 5$

$2y = 2$

$y = 1$

So the solution is $(3, 1)$.

✍️ Practice Quiz

Solve the following systems of equations using elimination by subtraction:

1. $4x + y = 9$

   $x + y = 3$

2. $5a + 2b = 13$

   $2a + 2b = 4$

3. $3m - n = 7$

   $m - n = 1$

💡 Conclusion

Elimination by subtraction is a powerful technique for solving systems of linear equations. By carefully subtracting equations, you can eliminate one variable and solve for the other. Remember to align your terms correctly and substitute back to find the complete solution!