Question

Solve the system of linear equations by graphing.\(-x+2 y=6\) \(x+4 y=24\)

Short answer

The solution to the system of equations is (4,5)

Step-by-step solution

Rewrite the equations in slope-intercept form (y = mx + b)

Equation 1: \(2y = x + 6\) which simplifies to \(y = 0.5x + 3\). Equations 2: \(4y = -x + 24\) which simplifies to \(y = -0.25x + 6\)

Graph the equations

Graph the two equations on the same set of axes. For each equation, plot the y-intercept (b) on the y-axis, then use the slope (m) to find the next point and draw the line. For Equation 1, the y-intercept is 3 and slope is 0.5. For Equation 2, the y-intercept is 6 and slope is -0.25.

Find the point of intersection

The point where the two lines intersect is the solution to the system of equations. By analysing the graph, it can be seen that the two lines intersect at (4,5).

Key concepts

Graphing Linear Equations

Graphing linear equations is an important skill when solving systems of equations. It lets us visually represent the solutions by drawing the lines that define each equation on a graph. Each linear equation can be represented as a straight line.

To graph a linear equation, follow these simple steps:

• First, make sure the equation is in the slope-intercept form, which is typically written as \(y = mx + b\).
• Identify the y-intercept (\(b\)), which is the point where the line crosses the y-axis.
• Plot the y-intercept on the graph.
• Use the slope (\(m\)) to determine the rise over run, meaning how much you go up or down with each step to the right.
• Draw a straight line through the plotted points to extend the graph of the equation.

By following these steps, you can graph multiple linear equations on the same set of axes to find where the lines meet, which helps in finding their intersection.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\). This form is quite useful as it provides important information about the line with just a glance.

Let's break down what each symbol means:

• \(m\) is the slope of the line. It tells us how steep the line is and the direction it leans. A positive slope means the line inclines upwards, while a negative slope indicates that it slopes downwards.
• \(b\) is the y-intercept, which represents the value of \(y\) when \(x\) is zero. This is where the line crosses the y-axis.

Knowing the slope \(m\) and the y-intercept \(b\) allows you to easily graph the line. Start plotting from the y-intercept and use the slope to determine other points along the line. This straightforward approach simplifies the graphing process and aids in visualizing solutions.

Intersection of Lines

The intersection of lines is a key concept in solving systems of linear equations. When you graph two or more linear equations, the point where they meet is called the intersection.

Here is why intersections are important when solving systems of equations:

• The intersection point represents the solution to the system, where all equations are satisfied simultaneously.
• This point can be identified by comparing the graphs to see where the lines cross.
• Sometimes lines might not intersect at all, indicating no solution, or they may overlap completely, indicating infinite solutions.

In the exercise, after graphing the equations \(y = 0.5x + 3\) and \(y = -0.25x + 6\), the lines intersect at the point (4,5). This point (4,5) is the solution to the system of equations, meaning both original equations are true when \(x = 4\) and \(y = 5\). Understanding how to find this intersection visually confirms the solution and strengthens comprehension of system behavior.