We are given the linear equation 2x plus 3 equals 5x minus 9. This is a linear equation with one variable, and our goal is to find the value of x that makes both sides equal.
Let's break down the equation. On the left side we have 2x plus 3, and on the right side we have 5x minus 9. The terms with x are called variable terms, and the numbers without x are constants. Our strategy will be to collect like terms together.
To solve for x, we first want to get all the x terms on one side. We'll subtract 2x from both sides. On the left, 2x minus 2x cancels out, leaving just 3. On the right, 5x minus 2x gives us 3x, so we have 3 equals 3x minus 9.
Now we want to get all the constant terms on the other side. We'll add 9 to both sides. On the left, 3 plus 9 gives us 12. On the right, minus 9 plus 9 cancels out, leaving just 3x. So we have 12 equals 3x.
To solve for x, we need to isolate it completely. Since x is multiplied by 3, we'll divide both sides by 3. On the left, 12 divided by 3 is 4. On the right, 3x divided by 3 gives us just x. Therefore, x equals 4.
Let's verify our solution by substituting x equals 4 back into the original equation. On the left side, we have 2 times 4 plus 3, which is 8 plus 3 equals 11. On the right side, we have 5 times 4 minus 9, which is 20 minus 9 equals 11. Since both sides equal 11, our solution x equals 4 is correct.
Let's summarize the complete solution process. We started with the original equation, then subtracted 2x from both sides, added 9 to both sides, and finally divided by 3 to get x equals 4. We verified our solution by substituting back into the original equation. The key principles we used were: maintaining equality by doing the same operation to both sides, combining like terms, and using inverse operations to isolate the variable.