91Ó°ÊÓ

Understanding the slope-intercept form is crucial when graphing systems of equations. The slope-intercept form of a linear equation is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is where the line crosses the y-axis.

Why is this form so helpful?

• Ease of Graphing: Knowing the slope \(m\) and the y-intercept \(b\) allows you to quickly plot two key points on the line: the y-intercept and another point obtained using the slope.
• Identifying Characteristics: Helps in easily identifying how steep the line is and its direction (increasing or decreasing).

In our step-by-step solution, we converted both given equations into slope-intercept form. The first equation \(3x - 5y = 7\) was transformed into \(y = \frac{3}{5}x - \frac{7}{5}\), giving us a slope of \(\frac{3}{5}\) and a y-intercept of \(-\frac{7}{5}\). Similarly, the second equation \(2x + y = 9\) became \(y = -2x + 9\), with a slope of \(-2\) and a y-intercept of \(9\). Understanding these components makes the graphing process intuitive.

The intersection point of two lines graphed on the coordinate plane is where the two lines meet. This point represents the solution to the system of equations, expressed as an ordered pair \((x, y)\).

Why is finding this intersection point important?

• Spotting Solutions: If two lines intersect, the coordinates of the intersection point are the solutions that satisfy both equations simultaneously.
• Visual Verification: Graphing provides a visual method to confirm solutions found through algebraic manipulation.

In our exercise, after transforming the equations into slope-intercept form and graphing the lines, we identified their intersection at the point \((3, -1)\). This demonstrates that substituting \(x = 3\) into either equation will produce \(y = -1\), confirming this point is the solution to the system.

In some cases, lines may be parallel (never intersecting) or the same line (infinitely many intersection points), each scenario providing different insights regarding the solutions.

Solving linear equations involves finding the values of variables that make the equations true. When dealing with systems of linear equations, it's about finding the point (or points) where those equations intersect. There are several methods to solve linear equations, but let's focus on using the information from our particular exercise:

To solve the system, we use the equality of slopes through setting equations equal to find \(x\). Once both equations were in slope-intercept form, we equated them:\[\frac{3}{5}x - \frac{7}{5} = -2x + 9\]Simplifying the equation:

• Get all terms involving \(x\) to one side, and constants to the other.
• Simplify to find \(x\). Here, \(x = 3\).
• Substitute \(x = 3\) back into one of the original equations to solve for \(y\), yielding \(y = -1\).

Thus the solution \((x, y) = (3, -1)\) signifies where both lines intersect. Understanding these steps makes solving linear equations less daunting by following a structured approach.