The equation y = 2x + 3 is in slope-intercept form, where the slope is 2 and the y-intercept is 3. This means the line crosses the y-axis at (0, 3) and for every 1 unit increase in x, y increases by 2.

In the slope-intercept form y = mx + b, the constant b represents the y-intercept. Here, b is 5, indicating the line crosses the y-axis at (0, 5).

To locate the intersection point, set the equations equal: x + 1 = -x + 3, which leads to x = 1 and subsequently y = 2. This confirms that the lines cross at (1, 2).

A unique solution occurs when the graphs of the equations intersect at exactly one point. This happens when the lines have different slopes, ensuring they cross each other only once.

The slope of a line is a measure of its steepness and direction. It indicates how much the y-coordinate changes for a unit change in the x-coordinate.

By solving x - y = 1 for x (x = y + 1) and substituting into 2x + 3y = 6, we find y = 4/5 and x = 9/5. This pair (9/5, 4/5) is the unique solution where both equations intersect.

When two lines have the same slope but different y-intercepts, they are parallel. Parallel lines do not meet, resulting in a system with no solution.

Simplifying 2y = 6x + 4 gives y = 3x + 2, which has the same slope as y = 3x - 2 but a different y-intercept. This indicates the lines are parallel and therefore do not intersect, resulting in no solution.

The elimination method works well with these equations because it allows one to cancel out a variable by adding or subtracting the equations. This makes the system easier to solve compared to the other methods listed.

Infinitely many solutions occur when both equations describe the same line. Essentially, the equations are equivalent, meaning every solution of one is also a solution of the other.

After solving the first equation for y and substituting into the second, the resulting identity indicates that both equations are equivalent. This means every solution of one equation satisfies the other, yielding infinitely many solutions.

Since the question specifies using a graph, the graphing method is the most straightforward approach. Plotting both lines lets you visually identify the intersection point.

A system with no solution is indicated by two parallel lines that do not meet at any point. This situation ensures that there is no coordinate pair that satisfies both equations simultaneously.

By substituting y = x - 1 into 2x + y = 7, we obtain 3x - 1 = 7, so x = 8/3 and y = 5/3. This unique solution represents where the two lines intersect.

The solution to a system of equations is the set of values that satisfies all the equations simultaneously. Graphically, this corresponds to the point or points where the graphs intersect.

First simplify 3(2x - y) = 12 to get 2x - y = 4, then express y as 2x - 4. Substituting into 4x + 5y = 7 yields x = 27/14 and y = -1/7, which is the intersection point of the two lines.

The second equation is exactly twice the first, which means both equations represent the same line. This dependency results in infinitely many solutions, making the system dependent.

For the system to have no solution, the lines must be parallel, meaning their slopes are equal but the y-intercepts differ. Setting the slopes equal leads to (k - 2)/4 = 2/(k + 1), which simplifies to k² - k - 10 = 0 with solutions k = (1 ± √41)/2.

Setting the x-components equal gives 1 + t = 1 + 2s, so t = 2s. Substituting t = 2s into the y-components, 2 - 3(2s) = 2 - s, leads to s = 0, and consequently t = 0. This yields the intersection point (1, 2).

For two lines to intersect at exactly one point, their slopes must be different. Since the second line has a slope of 2, choosing any value for m other than 2 will ensure the lines are not parallel and will have a unique intersection.