Question

Graph and check to solve the linear system. $$ \begin{aligned} &\frac{3}{4} x-\frac{1}{4} y=-\frac{1}{2}\\\ &\frac{1}{4} x-\frac{3}{4} y=\frac{3}{2} \end{aligned} $$

Short answer

The solution of the system of equations is the point (1,2).

Step-by-step solution

Graphing the Equations

First, let's start by graphing the two equations. To graph each line, you need to find two points that satisfy the equation and draw a line through them. For example, you can find the x and y intercepts. For the first equation \(\frac{3}{4} x-\frac{1}{4} y=-\frac{1}{2}\), when \(x=0\), \(y=2\), and when \(y=0\), \(x=-\frac{2}{3}\). Repeat the same step for the second equation and draw these two lines on a Cartesian plane.

Visually Identify the Intersection Point

The point of intersection of the two lines graphed in the Cartesian plane will give an estimate of the solution. This intersection point is the point that satisfies both of the equations simultaneously.

Algebraic Solution

To find the exact value of the intersection point, you can use substitution or elimination method. Here, we will use elimination method. Start by multiplying the first equation by 3 and the second by 4 and add them together this will eliminate \(y\), which leads to the equation \(x=1\). Substitute \(x=1\) in the first equation to get \(y=2\). So the solution to this system is (1,2).

Verification

Plug the values \(x=1\) and \(y=2\) into both equations to verify that they are satisfied. If both equations hold true, then the solution (1,2) is found to be correct.

Key concepts

Graphing Linear Equations

Understanding how to graph linear equations is fundamental in solving linear systems. To start graphing, you generally need two points. A convenient approach is to find where the line crosses the x-axis (x-intercept) and y-axis (y-intercept). In math terms, for the x-intercept, set the value of y to zero and solve for x, and vice versa for the y-intercept. With the given equations, you create a visual representation on the Cartesian plane. Graphing provides a graphical view of where the lines intersect, which corresponds to the solutions of the system if it exists.

Enhancing this concept for students begins with reinforcing how to find intercepts and plotting them correctly. Additionally, emphasizing the use of a straightedge ensures lines are drawn accurately for easier identification of the intersection point.

Intersection Point

The intersection point of two lines on a graph is the 'meeting point' and represents a set of coordinates that satisfy both equations. For example, with the equations provided, they intersect at the point (1,2). This point is significant as it's the solution to the linear system. To enhance understanding, students should be encouraged to double-check this point by substituting back into both equations.

Determining Intersection Accurately

Although visual inspection can give you an estimate, finding this point algebraically ensures the solution is precise. It's essential to explain that the graphical method is good for an estimate, but for precise solutions, algebra (like the elimination or substitution method) is necessary.

Elimination Method

The elimination method is one of the algebraic strategies for solving linear systems. It involves adding or subtracting the equations to eliminate one of the variables, leaving a single equation with one variable that can be solved easily. To utilize this method effectively:

• Both equations must be in standard form (Ax + By = C).
• If needed, multiply one or both equations by certain numbers to align the coefficients of one variable.
• Add or subtract the equations to eliminate that variable.

Once one variable is eliminated, find its value and substitute it back into one of the original equations to determine the other variable.

For students, explaining why and how coefficients are aligned and the importance of maintaining an equation's balance during manipulation is key to mastering the elimination method.

Algebraic Solution

An algebraic solution involves solving equations using algebraic techniques to find an exact answer. This approach is much more reliable than graphical methods as it provides an exact solution. After you've used the elimination method to find the value of one variable, you plug it back into any of the original equations to find the value of the other variable.

Practice Makes Perfect

Encourage lots of practice with different types of equations to become proficient with the algebraic solution process. To ensure clarity, emphasize the need to perform the same operations on both sides of the equation to maintain equality, which is a key concept of algebra. Understanding this principle is essential for students to solve linear systems successfully and independently.