/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Solve each equation for \(y\). ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 53

Solve each equation for \(y\). \(3 x+y=4\)

Short Answer

Expert verified
y = 4 - 3x

Step by step solution

01

- Isolate the term with y

Start by isolating the term that contains y on one side of the equation. Subtract 3x from both sides of the equation: 3x + y - 3x = 4 - 3x Which simplifies to: y = 4 - 3x

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Key Concepts

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

Isolation of Variables
When solving linear equations, isolating the variable you are solving for is a fundamental step. This means we rearrange the equation so that the variable is on one side and everything else is on the other. In our example, we are solving for y in the equation which is initially written as: 3x + y = 4. We must isolate y by subtracting 3x from both sides of the equation. This gives us: y = 4 - 3x. Now y is isolated, and we can see its relationship to x more clearly.

Remember:
  • Move all other terms away from the variable you're solving for.
  • Perform the same operation on both sides of the equation to maintain balance.
Linear Equations
Linear equations are equations between two variables that give a straight line when plotted on a graph. They have the form y = mx + b, where:
  • m is the slope of the line.
  • b is the y-intercept, where the line crosses the y-axis.
In the example y = 4 - 3x, comparing this to the standard form, we can identify that -3 is the slope (m), and 4 is the y-intercept (b). This means the line will cross the y-axis at y = 4 and it will have a negative slope, slanting downwards as we move from left to right.

Linear equations are simple yet powerful tools in algebra that describe a consistent relationship between two variables.
Algebraic Manipulation
Algebraic manipulation involves performing operations to both sides of an equation in order to solve for a variable or simplify the equation. Here are some basic rules:
  • Addition and subtraction: You can add or subtract the same number or expression from both sides.
  • Multiplication and division: You can multiply or divide both sides by the same non-zero number.
In about our example 3x + y = 4, we used subtraction to move 3x to the other side: subtract 3x from both sides which became y = 4 - 3x.

Algebraic techniques also include distributing, factoring, and combining like terms. The goal is always to simplify the equation to make it easier to solve for the desired variable. Practice these techniques to develop a strong foundation in algebra.

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Most popular questions from this chapter

The formula \(d=r t\) (distance \(=\) rate \(\times\) time) is used in the applications. If the rate of a boat in still water is \(10 \mathrm{mph}\), and the rate of the current of a river is \(x \mathrm{mph},\) what is the rate of the boat (a) going upstream (that is, against the current, which slows the boat down); (b) going downstream (that is, with the current, which speeds the boat up)?

34\. Explain one of the drawbacks of solving a system of equations graphically.

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Solve each problem. LeBron and Shaq measured a basketball court and found that the width of the court was \(44 \mathrm{ft}\) less than the length. If the perimeter was \(288 \mathrm{ft}\), what were the length and the width of the basketball court?

Give \((a)\) the additive inverse and (b) the multiplicative inverse (reciprocal) of each number. $$ \frac{7}{8} $$
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