Question

In the $xy$ -coordinate system, if $(m,n)$ and $(m+2,n+k)$ are two points on the line with the equation $x=2y+5,$ then $k=$ $$\begin{array}{l}\frac{1}{2}\\ \\ 1\\ 2\\ \frac{5}{2}\\ 4\end{array}$$

Short answer

k = 1

Step-by-step solution

- Understand the Line Equation

The equation of the line is given as $x=2y+5$. This equation shows the relationship between $x$ and $y$ coordinates of any point on this line.

- Plug in the First Point

First, substitute the coordinates of the point $(m,n)$ into the line equation. Since this point lies on the line, it must satisfy the equation: $$m=2n+5$$

- Plug in the Second Point

Next, substitute the coordinates of the point $(m+2,n+k)$ into the line equation. This point also lies on the line, so it must satisfy the same line equation: $$m+2=2(n+k)+5$$

- Set Up the Equation for the Second Point

From the equation $m+2=2(n+k)+5$, simplify to find $k$. Start by expanding and isolating $k$ on one side: $$m+2=2n+2k+5$$Subtract $2n+5$ from both sides: $$m+2-2n-5=2k$$ $$m-2n-3=2k$$

- Use the Equation from the First Point

Recall that from the first point, we have $m=2n+5$. Substitute this into the simplified equation from Step 4: $$(2n+5)-2n-3=2k$$ $$2n+5-2n-3=2k$$ $$2=2k$$ Divide both sides by 2: $$k=1$$

Key concepts

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality of two expressions. In our exercise, the equation given is a linear equation: $$x=2y+5$$. This means for any point $(x,y)$ on the line, the relationship between $x$ and $y$ satisfies this equation. Algebraic equations can be used to describe various relationships and solve problems if you correctly substitute values and follow algebraic principles. Here, we substitute the points $m,n$ and $m+2,n+k$ to form new equations. This helps us find unknown variables like $k$. Starting with basic substitutions and following through to simplify the equations, enables solving for $k$ effectively. Remember, understanding algebraic manipulation, such as isolating variables and simplifying expressions, is key to solving these problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses the coordinate plane to resolve geometric problems. By defining points through coordinates $(x,y)$, lines and curves can be represented with algebraic equations. Our line is represented as $$x=2y+5$$. To determine if points lie on this line, you substitute the coordinates into the equation.

• First point: Substituting $m,n$\ into $$x=2y+5$$, we get: $$m=2n+5$$.
• Second point: For $m+2,n+k$\, substituting into the equation gives $$m+2=2(n+k)+5$$.

These substitutions are crucial. They allow us to use the properties of the line to form equations that can be solved to find unknowns like $k$. The fundamental idea in coordinate geometry is expressing geometric situations in algebraic form for easier solving.

Linear Equations

Linear equations are equations of the first degree, involving constants and a single variable solved linearly. In our problem, the equation $$x=2y+5$$ depicts a line where $x$ changes twice as fast as $y$ plus a constant shift of 5. Linear equations can appear in different forms, like $y=mx+b$, but here it’s more direct, reflecting the relationship between x and y. To solve for $k$, the steps involve:

• Forming equations from known coordinates.
• Substituting values and simplifying to isolate the unknown variable.

The consistent addition, subtraction, multiplication, and division paths used in the step-by-step solution show how logically breaking down the equation makes it simpler. Mastering linear equations involves understanding these operations and how to manipulate the terms systematically.

Coordinate Systems

A coordinate system is a system that uses one or more numbers, or coordinates, to determine the position of points. In this exercise, we are dealing with the Cartesian coordinate system, where each point on a plane is identified by an $(x,y)$ pair. The relationship between these coordinates lets us graph lines and solve for intersections. Here’s how coordinates guide us to the solution:

• Points $m,n$\ and $m+2,n+k$\ relate to the line $x=2y+5$.
• To check if these points lie on the line, substitute their coordinates into the equation.
• From this, we form additional equations giving relationships between $m,n,$ and $k$.

The coordinate system allows systematically studying geometry through algebra, aiding in effectively solving problems like identifying specific values and verifying point positions on a line. Understanding this system is foundational in algebra, geometry, and even calculus.