We are given the linear equation 4x plus 3 equals 10x minus 9. This is a linear equation with one variable x, and our goal is to find the value of x that makes both sides of the equation equal.
To solve this equation, we first collect all terms with x on one side. We do this by subtracting 4x from both sides of the equation. On the left side, 4x minus 4x cancels out, leaving just 3. On the right side, 10x minus 4x gives us 6x, so we have 6x minus 9.
Now we need to isolate the term with x by moving the constant terms to the other side. We do this by adding 9 to both sides of the equation. On the left side, 3 plus 9 equals 12. On the right side, negative 9 plus 9 cancels out, leaving just 6x.
To find the value of x, we divide both sides of the equation by 6. On the left side, 12 divided by 6 equals 2. On the right side, 6x divided by 6 leaves just x. Therefore, x equals 2.
To verify our solution is correct, we substitute x equals 2 back into the original equation. For the left side: 4 times 2 plus 3 equals 8 plus 3 equals 11. For the right side: 10 times 2 minus 9 equals 20 minus 9 equals 11. Since both sides equal 11, our solution x equals 2 is correct.
Let's solve the equation using an alternative method by collecting x terms on the right side first. We subtract 10x from both sides. On the left side, 4x minus 10x gives us negative 6x plus 3. On the right side, 10x minus 10x cancels out, leaving negative 9. Next, we subtract 3 from both sides to isolate the x term, giving us negative 6x equals negative 12. Finally, we divide both sides by negative 6 to get x equals 2. This confirms our previous solution.
Let's summarize the key steps for solving linear equations like this one. First, collect like terms by moving all variable terms to one side and constants to the other. Second, isolate the variable term by performing inverse operations. Third, solve for the variable by dividing by its coefficient. Finally, verify your solution by substituting it back into the original equation. These steps form a general approach for solving any linear equation in one variable.