Problem 75 - Technical Mathematics with Calculus

Question

Solve for \(x.\) Simple Literal Equations. $$a x+4=b-3$$

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = \frac{b-7}{a}\).

Step 1: Isolate the Variable Term

Start by moving the constant on the same side as the variable term to the other side of the equation. This can be done by subtracting 4 from both sides of the equation to get the variable term by itself on one side. The equation becomes: \(ax = b - 3 - 4\).

Step 2: Combine Like Terms

Combine the constants on the right side of the equation. Subtract 3 from 4 to simplify the equation to \(ax = b - 7\).

Step 3: Divide by the Coefficient

To solve for the variable \(x\), divide both sides of the equation by the coefficient \(a\). This will isolate \(x\) on one side of the equation. The final step gives you \(x = \frac{b-7}{a}\).

Key Concepts

Isolate the Variable

One of the foundational techniques in algebra is learning how to isolate the variable. This means rearranging an equation so that the variable we are solving for is alone on one side of the equation. Isolation is the first critical step to solving literal equations. For instance, if we have an equation such as \( ax+4=b-3 \), our goal is to get \( x \) by itself.

To do this, we need to perform the same operation on both sides of the equation to maintain the balance—it's like keeping a seesaw level. We can subtract 4 from each side to move the constant away from the variable term, leading to \( ax = b - 3 - 4 \). This now readies us for the next important step: combining like terms.

Combine Like Terms

Once we've started the process of isolating the variable, we'll often encounter terms that can be simplified. This is where we combine like terms. In algebra, like terms are terms that have the same variable raised to the same power. Numbers without variables, known as constants, can also be like terms.

In our example with \( ax = b - 3 - 4 \), the numbers -3 and -4 are like terms because they are both constants. We combine them by simple addition or subtraction: -3 and -4 add together to give -7, which simplifies our equation to \( ax = b - 7 \). This act not only tidies up equations but also brings us one step closer to finding the value of our variable.

Coefficients in Algebra

In any algebraic expression, numbers that multiply a variable are called coefficients. For example, in the term \( ax \), \( a \) is the coefficient of \( x \). Understanding coefficients is crucial because it allows us to manipulate equations to solve for variables.

To solve for \( x \) in \( ax = b - 7 \), we need to divide both sides of the equation by the coefficient \( a \). This is how we 'cancel' the effect of the coefficient, leaving us with the isolated variable: \[ x = \frac{b-7}{a} \]. Don't forget that dividing by a coefficient is the same as multiplying by its inverse, enabling us to isolate the variable cleanly and solve the equation.