Problem 68 - Algebra 1

Question

Find the value of \(k\) so that the line through the given points has slope \(m .\) $$ (k+1, k-1),(k,-k) ; m=k+1 $$

Step-by-Step Solution

Answer

The value of \(k\) that makes the slope of the line equal to \(k+1\) is 0.

Step 1: Write Down Slope Formula

The basic formula calculating the slope of a line passing through two points \((x1, y1)\) and \((x2, y2)\) is given by \(m = (y2 - y1) / (x2 - x1)\).

Step 2: Substitution

Substitute the coordinates of the given points into the slope formula, resulting in \(m = (k+1 - (-k)) / ((k+1) - k)\).

Step 3: Simplify The Equation

After substituting, simplify the equation which gives \(m = 2k+1\).

Step 4: Solve The Equation

We already know that \(m = k + 1\), so replace \(m\) in the equation obtained in step 3, giving us \(k + 1 = 2k + 1\).

Step 5: Final Solution

Lastly, solve the equation for \(k\). Simplifying gives \(0 = k\).

Key Concepts

Slope Formula

The slope, often represented by the letter \(m\), is a measure of the steepness or incline of a line. It is calculated using the slope formula, which is based on two points on a line. The formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points. This formula calculates how much \(y\), the vertical change, grows for every single unit increase in \(x\), the horizontal change.

• If \(m > 0\), the line goes upwards from left to right.
• If \(m < 0\), the line goes downwards.
• If \(m = 0\), the line is horizontal.
• If \(m\) is undefined (division by zero), the line is vertical.

Understanding the slope helps in determining the nature of the relationship between \(x\) and \(y\). In this exercise, we substituted the given points into the formula to find the relationship between \(k\) and the slope \(m\).

Linear Equations

Linear equations are equations of the first order and form a straight line when graphed on a coordinate plane. The most common form of a linear equation in two variables is the slope-intercept form, given by: \[ y = mx + c \] where \(m\) is the slope and \(c\) is the y-intercept, the point where the line crosses the y-axis. Linear equations are crucial because they model such relationships in a simple, predictable form. For our exercise, we considered linear equations to solve for \(k\) because they provide a clear way to relate the changes in \(x\) and \(y\). In the solution, we matched the obtained expression for \(m\) from the points to the given \(m = k + 1\), leading us to a simple linear equation in terms of \(k\). Solving that linear equation gave us the result for \(k\).

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are solved together, as they have common solutions. They are often used when you have two or more equations working together to define relationships between variables. In this exercise, the form of simultaneous equations wasn't traditional but solving for \(k\) required adjusting to the information provided, like the slope equalling \(k + 1\), which acted like a secondary equation in our problem. We initially simplified the equation derived from the slope formula to express \(m\) in terms of \(k\) and then used the condition given for \(m\). This was like solving two equations simultaneously. Though there wasn't an explicit set of equations, the logical setup implied working with an equivalent of simultaneous ideas, which resulted in precisely determining the value of \(k\).