Question

Find the value of \(y\) so that the line passing through the two points has the given slope. $$(0, y),(2,5), m=2$$

Short answer

The value of \(y\) that results in a slope of 2 for the line passing through the points \((0, y)\) and \((2,5)\) is \(y = 1\).

Step-by-step solution

Use the Slope Formula

The formula for the slope of a line \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \[\ m = \frac{y_2 - y_1}{x_2 - x_1} \] In the given exercise, \((x_1, y_1) = (0, y)\) and \((x_2, y_2) = (2,5)\), and \(m = 2\). Substituting these into the slope formula gives \[\ 2 = \frac{5 - y}{2 - 0} = \frac{5 - y}{2} \]

Solve for y

To find the value of \(y\), start by multiplying both sides of the equation by 2: \[ \ 2 \times 2 = \frac{5 - y}{2} \times 2 \ \] This results in: \[ \ 4 = 5 - y \ \] Now, add \(y\) and subtract 4 from both sides of the equation to solve for \(y\): \[ y = 5 - 4 \]

Compute the Value of y

The final step is to compute the value of \(y\). After performing the subtraction, the result is: \[ \ y = 1 \]

Key concepts

Slope Formula

Understanding the slope formula is crucial in coordinate geometry and algebra. The slope of a line measures its steepness and direction, essentially describing how much the line rises or falls over a horizontal distance. The slope formula is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) is the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line.

To calculate the slope using this formula, subtract the \( y \) value of the first point from the \( y \) value of the second point and divide the result by the subtraction of the \( x \) value of the first point from the \( x \) value of the second point. This can also be seen as the 'rise' over the 'run'. In practical problems, like the one given in the exercise, you use this formula to find an unknown coordinate when the slope and one point are known, as well as when determining the slope between two known points.

Solving Linear Equations

When you're faced with a linear equation, the goal usually is to solve for the unknown variable. This process involves performing operations that will get the variable on one side of the equation, leaving a numerical value on the other side. These operations must preserve the equality, which means whatever you do to one side of the equation, you must do to the other side.

In the step-by-step solution for our exercise, to find the value of \( y \) that makes the slope equal to 2, we multiply both sides by 2 to eliminate the fraction, and then isolate \( y \) by moving terms from one side of the equation to the other. This is a classic example of using inverse operations: multiplication to clear a fraction, subtraction to isolate the variable, and tweaking to achieve a simplified form of the equation - all of which are fundamental to solving linear equations effectively.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method combines algebra and geometry to describe the position of points, lines, and figures in space using coordinates - usually in 2 or 3 dimensions. In problems like the one presented, we work with the Cartesian coordinate system, which uses two perpendicular axes, commonly identified as \( x \) and \( y \) axes.

In this context, finding the slope is one of the essential concepts, as it enables us to define the orientation of lines. Furthermore, the understanding of coordinate geometry is pivotal when we calculate distances between points, define equations for specific geometric figures, and explore the relationship between algebraic equations and geometric curves. The ability to analyze and solve problems within this coordinate plane is a foundational skill for further studies in both geometry and algebra.