Problem 59 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ -2 x+11 y=14 \quad \text { for } x $$

Step-by-Step Solution

Answer

\( x = -7 + \frac{11y}{2} \)

Step 1: Identify the Equation to Solve

The equation given is \( -2x + 11y = 14 \), and we are asked to solve for \( x \). This means we need to manipulate the equation so that \( x \) is isolated on one side.

Step 2: Add or Subtract Terms to Isolate Terms with x

To isolate the terms with \( x \), we need to move the \( 11y \) term to the right side of the equation. We can do this by subtracting \( 11y \) from both sides.\[-2x + 11y - 11y = 14 - 11y\]This simplifies to:\[-2x = 14 - 11y\]

Step 3: Divide by the Coefficient of x

Now, to solve for \( x \), we need to divide each term in the equation by \(-2\), which is the coefficient of \( x \).\[x = \frac{14 - 11y}{-2}\]This can be further simplified to:\[x = -\frac{14}{2} + \frac{11y}{2}\]\[x = -7 + \frac{11y}{2}\]

Key Concepts

Solving Equations

When we talk about solving equations, we're essentially trying to find the value of the variable that makes the equation true. In this exercise, we had the equation \(-2x + 11y = 14\). To solve an equation, you often perform a series of steps that simplify the expression. This usually involves isolating the variable you're solving for by performing operations like addition, subtraction, multiplication, or division on both sides of the equation. This ensures the equation remains balanced.The goal is to have the variable (in this case, \(x\)) by itself on one side of the equation, leading us to its value. Remember:

• Keep the equation balanced.
• Alter both sides of the equation in the same way.
• Acknowledge the operations that have inverse actions, like addition and subtraction, or multiplication and division, to help simplify and solve.

Through these methods, solving becomes a systematic process, guiding you step-by-step.

Isolation of Variables

Isolating a variable means rearranging the equation so that the variable you're interested in is alone on one side of the equation. In the given problem, we were asked to isolate \(x\) in the equation \(-2x + 11y = 14\).To begin, we looked at ways to remove any other terms or coefficients that are affecting \(x\). Subtraction and addition are common tools for moving terms across the equals sign. We subtracted \(11y\) from both sides, leading us to \(-2x = 14 - 11y\).Next comes the division, where we divide by the coefficient of \(x\), which is \(-2\), to achieve our goal of having \(x\) alone. This gave us:

• \(x = \frac{14 - 11y}{-2}\)
• Simplifies to: \(x = -7 + \frac{11y}{2}\)

The above procedure is crucial because isolating variables allows us to see directly how each variable relates to the others in the equation.

Linear Equations

Linear equations are equations between two variables that produce a straight line when plotted on a graph. The standard form is typically \(Ax + By = C\). In this exercise, our given equation, \(-2x + 11y = 14\), is already in linear form.Such equations express a linear relationship, meaning every increase in one variable results in a proportional change in the other. Linear equations are fundamental in algebra because they are straightforward to manipulate and provide a foundation for more complex equation solving.Knowing their structure helps in understanding the relationship between variables, making it easier to predict dynamics represented by the equation. You can expect:

• Consistent rates of change.
• One solution if plotted with infinite intersections.
• Easy application of algebraic operations to isolate variables.

Grasping linear equations is pivotal in mastering algebra, as they represent foundational links between various quantities and are often the first step to comprehending more intricate mathematical concepts.