Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system. $$ \left\\{\begin{array}{l} 5 x-y=21 \\ 2 x+3 y=-12 \end{array}\right. $$

Short answer

The solution is \((3, -6)\). The system is consistent.

Step-by-step solution

Express one variable in terms of the other

Solve the first equation for one of the variables. From the equation \(5x - y = 21\), isolate \(y\):\[y = 5x - 21\]

Substitute and solve for one variable

Substitute \(y\) in the second equation with the expression obtained in Step 1:\[2x + 3(5x - 21) = -12\]Simplify and solve for \(x\):\[2x + 15x - 63 = -12\]\[17x = 51\]\[x = 3\]

Solve for the other variable

Substitute \(x = 3\) back into the expression for \(y\) from Step 1: \[y = 5(3) - 21\]\[y = 15 - 21\]\[y = -6\]

Check the solution

Substitute \(x = 3\) and \(y = -6\) back into the original equations to verify the solution:First equation: \[5(3) - (-6) = 21\]\[15 + 6 = 21\] (True)Second equation: \[2(3) + 3(-6) = -12\]\[6 - 18 = -12\] (True)

Graph the lines

Graph the lines of the system by plotting the equations:For \(5x - y = 21\), intercepts are (0, -21) and (21/5, 0).For \(2x + 3y = -12\), intercepts are (0, -4) and (-6, 0).Plot these points and draw the lines. The intersection point \((3, -6)\) is the solution.

Key concepts

solving linear systems

Solving linear systems involves finding the point where two or more linear equations intersect. This point represents the values of the variables that satisfy all the equations simultaneously. In our exercise, we have two equations: \(5x - y = 21\) and \(2x + 3y = -12\). To solve this system, we follow several steps, such as expressing one variable in terms of another, substituting, and solving for each variable. After solving, it's crucial to verify the solution by substituting back into the original equations. This confirms that the solution is correct.

graphing lines

Graphing lines helps visualize the solution of a linear system. Each line in the graph represents an equation in the system. To graph equations like \(5x - y = 21\) and \(2x + 3y = -12\), it’s essential to find their intercepts.

Here are the key steps to graph these lines:

• Find x-intercept (where y = 0) and y-intercept (where x = 0) for each equation.
• Plot these intercepts on the coordinate plane.
• Draw the lines through these points.

For the given example, after plotting the points (0, -21) and (4.2, 0) for the first equation, and (0, -4) and (-6, 0) for the second equation, you can draw their lines and identify the intersection point, (3, -6), which is the solution.

substitution method

The substitution method is a way to solve systems of equations algebraically. This method involves three main steps:

• Isolate one of the variables in one of the equations. For example, from \(5x - y = 21\), we get \(y = 5x - 21\).
• Substitute this expression into the other equation to solve for the remaining variable. Substitute \(y = 5x - 21\) into \(2x + 3y = -12\), giving \(2x + 3(5x - 21) = -12\).
• Solve the resulting equation for one variable and then use this value to find the other variable. In our case, solving gives \(x = 3\), which is then substituted back to find \(y = -6\).

This method is efficient and clear, especially when one variable can be easily isolated.

inconsistent systems

Not all systems of linear equations have a solution. Some systems are inconsistent, meaning they have no solutions because the lines are parallel and never intersect. An inconsistent system is identified when, after attempting to solve the equations, you end up with a contradiction such as a statement where equals do not match (like \(0 = 5\)).

For example, suppose after substituting, you get equations indicating a paradox (e.g.,\(5 = -3\)). This would mean the lines never meet and the system is inconsistent. However, in the given example, the system of equations \(5x - y = 21\) and \(2x + 3y = -12\) is consistent, since they intersect at (3, -6) indicating a valid solution.