Problem 23 - Algebra 1: Concepts and Skills

Question

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{aligned} &y=-2 x+6\\\ &y=2 x+2 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1\) and \(y = 4\).

Step 1: Graphical Solution

Plot each equation on the graph. The \(y=-2x+6\) will be a downtrending line starting from the point (0,6) on the y-axis. The \(y=2x+2\) will be an uptrending line starting from the point (0,2) on the y-axis. The point where these two lines intersect will be the solution of the system.

Step 2: Estimate intersection point

By visual estimation, determine the coordinates of the intersection point, which represents the x and y solutions of the system. If drawn accurately, the lines should intersect at the point (1,4).

Step 3: Algebraic Solution

Set the two equations equal to each other and solve for x. So,-2x + 6 = 2x + 2. Solving this equation gives x = 1. Substitute x = 1 into either of the original equations to solve for y. Both will give y = 4.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is foundational to algebra. A linear equation represents a constant change and is graphically shown as a straight line on a coordinate plane. Each linear equation has the form of \(y = mx + b\), where \(m\) represents the slope, indicating the steepness of the line, and \(b\) is the y-intercept, which is where the line crosses the y-axis.

To graph linear equations, such as \(y=-2x+6\) and \(y=2x+2\), follow these steps:

• Identify the y-intercept (\(b\)). For the first equation, the y-intercept is 6, which means the line crosses the y-axis at the point (0,6). For the second, it is 2, crossing at (0,2).
• Plot the y-intercept on the graph.
• Determine the slope (\(m\)). A slope of -2 means that for every 1 unit you move right on the x-axis, you move 2 units down; a slope of 2 means you move 2 units up.
• From the y-intercept, use the slope to find another point on the line.
• Draw a straight line through the points. This represents the equation.

By graphing two or more linear equations on the same axes, you can visually identify points of intersection which represent the solutions to the system of equations.

Algebraic Solution of Linear Systems

An algebraic solution of linear systems offers a precise method for finding where two equations intersect, which means solving for the exact values of x and y that satisfy both equations simultaneously. This process typically involves substitution or elimination methods; however, in this exercise, the equations are set equal to each other since they both solve for y.

Here's how you proceed algebraically:

• Set the two equations equal to each other (since they both equal y), giving us \(-2x + 6 = 2x + 2\).
• Rearrange the equation to solve for x. In this case, by adding 2x to both sides and subtracting 2 from both sides, you isolate x and find \(x = 1\).
• With the value of x known, substitute it back into either original equation to solve for y. Both equations will give the same result: \(y = 4\).
• Confirm that the solution \((1,4)\) exists on both original lines, which means it is the point of intersection.

An experienced insight into solving mathematically provides accuracy and confirms the estimations made from the graphs.

Intersection Point Estimation

Intersection point estimation is a graphical technique used to approximate the point at which two or more lines on a graph meet. It becomes especially handy when you're not able to solve the system algebraically at the moment or when an estimate is sufficient for the task at hand.

The estimation involves a keen eye and can be supported by the following steps:

• After both lines are graphed, visually locate the area where they cross. Accuracy improves as the lines are drawn more precisely.
• Determine the approximate coordinates of this intersecting point by seeing which grid lines it is closest to.
• It is essential to cross-check this estimation with the algebraic solution to ensure accuracy when required.

For the example equations, a well-drawn graph reveals that the lines intersect near the point (1,4), which acts as a helpful prediction before proceeding to a more rigorous algebraic solution.