Question

If the equations of two lines are given by \(y=-28 x-4\) and \(y=2 x+11\), then at which of the following points do the two lines intersect? A. \(\left(-\frac{9}{2}, 2\right)\) B. \((2,15)\) C. \(\left(-\frac{1}{2}, 10\right)\) D. \((-26,7)\)

Short answer

The point of intersection of the two lines is \(\left(-\frac{1}{2}, 10\right)\). So, the correct answer is C.

Step-by-step solution

Write down the given equations

The given equations are: 1) \(y = -28x - 4\) 2) \(y = 2x + 11\)

Solve the system of equations

Since both equations are already solved for y, we can equate the right-hand sides and solve for x: \(-28x - 4 = 2x + 11\)

Solve for x

Add 28x to both sides and subtract 11 to isolate x: \(-4 - 11 = 2x + 28x\)

Simplify and solve for x

Simplify the equation: \(-15 = 30x\) Divide both sides by 30: \(x = -\frac{1}{2}\)

Substitute x value into one of the equations to find y

We can use either equation to find y, but we'll use the second equation because it has smaller coefficients: \(y = 2\left(-\frac{1}{2}\right) + 11\)

Solve for y

Simplify the equation and solve for y: \(y = -1 + 11\) \(y = 10\)

Identify the point of intersection

The point of intersection is \(\left(-\frac{1}{2}, 10\right)\), which matches option C. So, the correct answer is C.

Key concepts

Linear Equations Intersection

The intersection of linear equations is a fundamental concept in algebra that refers to the point where two lines cross on a graph. When two lines intersect, it means that they share a common point, which corresponds to a set of x and y values that satisfy both equations simultaneously.

Let's visualize it with a real-life example. Imagine two roads crossing each other at a traffic junction. Each road can be represented by a linear equation, and the intersection point of these roads is similar to the solution we are looking for in our algebra problem.

By finding the x-value where the two equations intersect, we can then substitute it back into one of the original equations to solve for the y-value. The result gives us the exact location of the intersection point on the coordinate plane. This intersection point is significant because it provides a common solution that applies to both linear equations involved in the problem.

Algebra Problem Solving

Algebra problem solving is all about finding ways to manipulate and solve for unknown variables within equations and expressions. It's akin to piecing together a puzzle where each step brings you closer to seeing the full picture. In the realm of linear equations, this often involves finding the value of variable 'x' that can then be used to uncover 'y'.

To successfully navigate through an algebra problem, it is critical to comprehend the properties of the operations used, such as addition, subtraction, multiplication, and division. The key is maintaining equality on both sides of the equation as you perform operations to isolate the variable. Consider the order of operations, and make strategic moves to simplify the equation as efficiently as possible. Clear strategies, like combining like terms and balancing equations, are the stepping stones to achieving solutions.

The elegant dance of algebra problem solving thus lies in making the right moves at the right time, all while keeping the scales of our algebraic balance equal.

Equation Solving Steps

The steps involved in solving equations serve as a critical pathway to unveiling unknown values within the equations. The process typically follows a structured approach that helps keep the methodology clear and organized.

Here's a simplified breakdown of this process:

• Step 1: Write down the given equations clearly.
• Step 2: If necessary, manipulate the equations to isolate the variables, either by solving for y or by setting the equations equal to each other if they have a common variable.
• Step 3: Simplify the equation step by step, combining like terms and reducing complexity.
• Step 4: Perform operations to isolate the unknown variable on one side of the equation.
• Step 5: Once the value of one variable is found, substitute this value back into one of the original equations to find the other variable.
• Step 6: Cross-check the solution in both equations to ensure accuracy.
• Step 7: Present the solution as an ordered pair that represents the intersection point of the two lines.

Following these systematic steps ensures that each part of the equation is addressed and that the path to the solution is as clear as possible.