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Chapter 2: Problem 37

Solve each formula for the indicated variable. \(3 x+2 y=5\) for \(y\)

Short Answer

Expert verified
\(y = \frac{5 - 3x}{2}\)

Step by step solution

01

Isolate on one side of the equation

We first need to isolate \(y\) on one side of the equation. To do this, remove any terms not containing \(y\) from the left side of the equation. We can subtract \(3x\) from both sides of the equation: \(2y = 5 - 3x\).
02

Solve for \(y\)

To get \(y\) alone, we then need to remove the coefficient \(2\) in front of \(y\). We can do this by dividing the entire equation by 2: \(y = \frac{5 - 3x}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are mathematical expressions that form a straight line when graphed. They generally have the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. These equations represent relationships between two variables.
Linear equations can be powerfully simple. With just two basic operations - addition (or subtraction) and multiplication (or division) - they hold a fundamental place in algebra.
It's important to remember that the solutions to a linear equation can tell us how changing one variable affects another. In practical terms, they're commonly used in plotting graphs and finding intersections of lines.
Solving for a Variable
Solving for a variable involves isolating one of the variables on one side of the equation so we can see how it relates to the other terms in the equation. By doing this, we learn more about the connections within the equation.
In our example, we solved for \(y\) in the equation \(3x + 2y = 5\). This required getting \(y\) by itself on one side of the equation.
  • The first thing we did was move the term containing \(x\) to the other side. This action set the stage for isolating \(y\).
  • Next, we dealt with the coefficient of \(y\), the number multiplying it. We divided both sides by 2 to completely isolate \(y\).

Solving for a variable allows us to express one variable entirely in terms of other variables or constants. It helps in various fields, from mathematics and science to everyday decision-making.
Equation Rearrangement
Equation rearrangement is the art of transforming an equation to a more useful form without changing its solutions. It involves using properties of equality - adding, subtracting, multiplying, or dividing both sides by the same number or expression.
In our step-by-step solution, equation rearrangement was used to make \(y\) the subject of the formula.
  • By subtracting \(3x\) from both sides of the equation, we removed \(x\)-related terms from \(y\)'s side.
  • Then, dividing both sides by 2 isolated \(y\), presenting \(y\) in terms of \(x\).

Rearranging equations helps solve real-world problems where a certain variable needs to be known explicitly. It鈥檚 a crucial step for understanding relationships within an equation or preparing it for further computations.

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Most popular questions from this chapter

Explain what it means for a number to satisfy an equation.

Solve each inequality and graph the solution on the number line. $$0 < 10-5 x \leq 15$$

Explain why multiplying both sides of an inequality by a negative constant reverses the direction of the inequality.

When a car of mass \(m\) collides with a wall, the energy of the collision is given by the formula \(E=\frac{1}{2} m v^{2} .\) Compare the energy of two collisions: a car striking a wall at \(30 \mathrm{mph}\), and at \(60 \mathrm{mph}\).

Express each solution as an inequality. Geometry A rectangle's length is 3 feet less than twice its width, and its perimeter is between 24 and 48 feet. What might be its width?
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