Question

A point on the line \(4 x+3 y=b\) is \((3,2)\). What is the \(y\) coordinate for \(x=4\) ? A. 3 B. \(\frac{3}{2}\) C. \(\frac{2}{3}\) D. -3

Short answer

The y coordinate when x = 4 is \(\frac{2}{3}\), which is answer C.

Step-by-step solution

Identify the given point

Identify that the given point on the line is \(3, 2\). This means when \(x = 3\) and \(y = 2\), the equation \(4x + 3y = b\) holds true.

Substitute the given point into the equation

Substitute \(x = 3\) and \(y = 2\) into the equation \(4x + 3y = b\). This results in: \[4(3) + 3(2) = b\]

Solve for b

Calculate the value of \(b\): \[4(3) + 3(2) = 12 + 6 = 18\] So, \(b = 18\).

Substitute x into the equation

Substitute \(x = 4\) and \(b = 18\) back into the equation \(4x + 3y = b\): \[4(4) + 3y = 18\]

Solve for y

Isolate \(y\) in the equation: \[16 + 3y = 18\] Subtract 16 from both sides: \[3y = 2\] Divide both sides by 3: \[y = \frac{2}{3}\]

Identify the correct answer

The \(y\) coordinate when \(x = 4\) is \(\frac{2}{3}\). So, the correct answer is \(\text{C}\).

Key concepts

Linear Equations

Linear equations are equations of the first degree, which means they graph as straight lines. A common form of linear equations is the slope-intercept form, written as \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. In this exercise, the equation \( 4x + 3y = b \) is in standard form. Standard form linear equations can still be manipulated to find values of \( x \) and \( y \).

To solve these equations, you often need to rearrange them to isolate one variable at a time. For example, our exercise asked to find the value of \( y \) when \( x = 4 \). After substituting and isolating \( y \), we end up solving a simple linear equation. These fundamental steps are essential in mastering linear equations.

Coordinate Geometry

Coordinate geometry uses algebra to study geometric problems by placing them in a coordinate system. Points in this system are identified by their \( (x, y) \) coordinates. The given exercise involves a point \( (3, 2) \) on the line defined by the equation \( 4x + 3y = b \). Understanding how these points fit into the equation is crucial.

When a point satisfies an equation of a line, it means that substituting the \( x \) and \( y \) values of the point into the equation makes it a true statement. This can be useful to find unknowns, such as the parameter \( b \) in our problem.

Once you know the value of \( b \), you can use the equation to find other points on the line. This involves solving for one variable when another is given, as seen when we found \( y \) for \( x = 4 \).

Solving for Variables

Solving for variables is a key skill in algebra. It involves isolating the variable on one side of the equation to determine its value. Let's go through our exercise step-by-step to illustrate this process. Starting with the equation \( 4x + 3y = b \), we first substituted the given point \( (3, 2) \). This enabled us to find \( b \). After substituting \( x = 4 \), we isolated the variable \( y \) with basic algebraic manipulations:

- Substitute known values: Given new values or already known variables into the equation.
- Simplify the equation: Carry out basic arithmetic operations to combine like terms and simplify the equation as much as possible.
- Isolate the variable: Use operations like addition, subtraction, multiplication, and division to get your desired variable by itself on one side of the equation.

In our problem, the steps led us to find \( y = \frac{2}{3} \) after replacing \( x \) with 4 and subtracting and dividing accordingly. Practicing these steps with different problems hones your problem-solving skills significantly.