Question

At what point in the \(x y\) -plane will the functions \(y=4 x-3\) and \(y=-\frac{1}{2} x+\) 2 intersect? (A) \(\left(2,-\frac{2}{3}\right)\) (B) \(\left(-\frac{3}{4}, \frac{5}{6}\right)\) (C) \(\left(\frac{10}{9}, \frac{13}{9}\right)\) (D) \(\left(1, \frac{3}{7}\right)\)

Short answer

The functions intersect at \(\left(\frac{10}{9}, \frac{13}{9}\right)\), option (C).

Step-by-step solution

Equating the two functions

To find the intersection point of the two functions, set the equations equal to each other: \[ 4x - 3 = -\frac{1}{2}x + 2 \]

Collect terms

Add \(\frac{1}{2}x\) to both sides to move all terms involving \(x\) to one side: \[ 4x + \frac{1}{2}x - 3 = 2 \] Combine the \(x\)-terms:\[ \frac{9}{2}x - 3 = 2 \]

Solve for \(x\)

Add 3 to both sides to isolate the \(x\)-term:\[ \frac{9}{2}x = 5 \] Multiply both sides by \(\frac{2}{9}\) to solve for \(x\):\[ x = \frac{10}{9} \]

Substitute \(x\) back into one equation

Use \(y = 4x - 3\) to find \(y\). Substitute \(x = \frac{10}{9}\): \[ y = 4\left(\frac{10}{9}\right) - 3 \]Simplify:\[ y = \frac{40}{9} - \frac{27}{9} = \frac{13}{9} \]

Identify the intersection point

The functions intersect at the point \( \left( \frac{10}{9}, \frac{13}{9} \right) \). Check this matches the provided options.

Key concepts

System of Equations

When dealing with algebra problems like finding where two lines intersect, solving a system of equations becomes vital. A system of equations involves finding common solutions for two or more equations. In this context, the two equations represent lines in the coordinate plane, defined by their respective linear equations.

• A system of equations can be solved using various methods, such as substitution, elimination, or graphing. Here, substitution is a handy choice.
• By setting the two equations equal, we solve for one variable. This essentially means finding a value for the variable that satisfies both equations simultaneously.
• In our original exercise, we equate the equations: \[ 4x - 3 = -\frac{1}{2}x + 2 \]
• Solving this equation helps us identify the x-coordinate of the intersection point.

Understanding how to solve systems of equations is an essential skill because it allows us to explore where two or more conditions are in sync in practical scenarios.

Coordinate Geometry

Coordinate geometry is a powerful mathematical tool that helps us describe and analyze the properties of geometric figures using a coordinate system. This system involves the use of two axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). Here's why coordinate geometry matters when solving intersection problems:

• The x-axis and y-axis serve as a reference to locate points within the plane using ordered pairs \((x, y)\).
• Linear equations can express lines in this plane, giving us a visual insight into their slope and position.
• For example, the line equation \(y = 4x - 3\) has a slope of 4, indicating it rises quickly as x increases, and intercepts the y-axis at \(-3\).
• Similarly, the equation \(y = -\frac{1}{2} x + 2\) shows a gentler downward slope with a positive y-intercept.

These insights into line characteristics help us anticipate where lines could intersect and guide us in checking our algebraic solutions visually.

Intersection Point

Finding the intersection point of two lines is a common problem in algebra and coordinate geometry. It can be broadly defined as the point where two lines cross each other in the coordinate plane. Here's how to approach finding such a point:

• First, solve a system of equations representing the two lines to find their intersection point.
• Once the x-value from the system is identified, substitute it back into one of the original equations to solve for y.
• For instance, substituting \(x = \frac{10}{9}\) into \(y = 4x - 3\) helped us calculate \(y = \frac{13}{9}\).
• The intersection point is thus represented as an ordered pair \(\left( \frac{10}{9}, \frac{13}{9} \right)\).

Finding intersection points is not only about calculation but also about comprehending how different lines or conditions might concurrently meet within a system. This understanding is useful in real-world contexts, such as determining meeting points in different routes or reconciling simultaneous constraints.