Exercises 143–145 will help you prepare for the material covered in the next section. Solve for y: 3x + 2y − 4 = 0.

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Rewrite the equation to isolate the term containing y. Start by subtracting 3x and adding 4 to both sides of the equation: 3x + 2y - 4 = 0 becomes 2y = -3x + 4.

Divide every term in the equation by 2 to solve for y. This gives y = (-3x/2) + (4/2).

Simplify the fractions in the equation. The term -3x/2 remains as is, and 4/2 simplifies to 2. Thus, the equation becomes y = (-3x/2) + 2.

Interpret the result as the slope-intercept form of a linear equation, y = mx + b, where m is the slope (-3/2) and b is the y-intercept (2).

Verify your work by substituting the expression for y back into the original equation to ensure it satisfies 3x + 2y - 4 = 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The general form is Ax + By + C = 0, where A, B, and C are constants. Understanding linear equations is crucial for solving for a specific variable, as they represent straight lines when graphed.

Isolating Variables

Isolating a variable involves rearranging an equation to solve for one variable in terms of others. This process often includes adding, subtracting, multiplying, or dividing both sides of the equation by the same number. Mastery of this technique is essential for solving equations like 3x + 2y - 4 = 0, where we need to express y in terms of x.

Graphing Linear Equations

Graphing linear equations involves plotting points that satisfy the equation on a coordinate plane, resulting in a straight line. The slope-intercept form, y = mx + b, is particularly useful for identifying the slope and y-intercept. Understanding how to graph these equations helps visualize the relationship between variables and verify solutions.