Problem 46 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} 5 x+2 y=7 \\ -10 x-4 y=-14 \end{array}\right. $$

Step-by-Step Solution

Answer

Infinitely many solutions.

Step 1: Rewrite Each Equation in Slope-Intercept Form

Rewrite the given system of equations in the form of \( y = mx + b \). The first equation is \( 5x + 2y = 7 \). Subtract \( 5x \) from both sides to get \( 2y = -5x + 7 \), then divide every term by 2: \( y = -\frac{5}{2}x + \frac{7}{2} \). The second equation is \( -10x - 4y = -14 \). Add \( 10x \) to both sides to get \( -4y = 10x - 14 \), then divide every term by -4: \( y = -\frac{10}{4}x + \frac{14}{4} \), and simplify to \( y = -\frac{5}{2}x + \frac{7}{2} \).

Step 2: Identify the Lines and Plot Them

Notice that both equations are actually the same (\( y = -\frac{5}{2}x + \frac{7}{2} \)). This means they represent the same line. To plot this line, find two points. For instance, when \( x = 0 \), \( y = \frac{7}{2} \) (3.5), and when \( x = 2 \), \( y = -3 \). Plot these points on the coordinate grid and draw the line through them.

Step 3: Determine the Solution

Since both equations simplify to the same line, they intersect at every point on this line. This means they have infinitely many solutions. Every point on the line \( y = -\frac{5}{2}x + \frac{7}{2} \) is a solution to the system.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a key process in understanding how to visually represent algebraic equations on a coordinate plane. To start, we need to rewrite the equations in a form that makes graphing straightforward. This is typically done using the slope-intercept form, which allows us to easily identify the slope and y-intercept.

Once the equations are rewritten, plot the lines on the coordinate grid. Each line corresponds to an equation from the system. The points where these lines intersect, if any, represent the solutions to the system.

In our example, after rewriting the given system of equations, we noticed that both equations represent the same line. This tells us that instead of intersecting at a single point, the two lines coincide with each other.

Slope-Intercept Form

Slope-intercept form is a way of writing linear equations so that we can easily identify their slope and y-intercept. The standard form is written as:

\( y = mx + b \)

Here, the coefficient \( m \) represents the slope, and the constant \( b \) represents the y-intercept. The slope indicates how steep the line is and in which direction it tilts. The y-intercept is the point where the line crosses the y-axis.

Let's apply this to our example:

• Starting with the first equation: \( 5x + 2y = 7 \). Subtract \( 5x \) from both sides: \( 2y = -5x + 7 \). Divide by 2 to get: \( y = -\frac{5}{2}x + \frac{7}{2} \).
• The second equation: \( -10x - 4y = -14 \). Add \( 10x \): \( -4y = 10x - 14 \). Divide by -4: \( y = -\frac{5}{2}x + \frac{7}{2} \).

In both cases, we ended up with the same slope-intercept form!

Infinitely Many Solutions

A system of linear equations can have different types of solutions:

• No solution: The lines are parallel and never intersect.
• One solution: The lines intersect at exactly one point.
• Infinitely many solutions: The lines overlap completely.

In our example, the system has infinitely many solutions because both equations are identical once simplified.

This means every point on the line \( y = -\frac{5}{2}x + \frac{7}{2} \) is a solution to both equations. When graphed, the lines would coincide, indicating they occupy the same space on the plane, thus giving us an infinite number of intersection points.