Similarly, for a system of linear equations in two variables, the unique solution is an ordered pair (x, y) which will satisfy both the equations in the system. Let understand the concept of a unique solution using a linear equation in one variable, 4x = 8 has a unique solution x = 2 for which the L.H.S is equal to the R.H.S. The unique solution of a system of equations means that there exists only one value for the variable or the point of intersection of the lines representing those equations, on substituting which, L.H.S and R.H.S of all the given equations in the system become equal.įor example, we know that a linear equation in one variable will always have one solution. There can be different types of solutions to a given system of equations, The main reason behind solving an equation system is to find the value of the variable that satisfies the condition of all the given equations true. We compute the values of the unknown variables still balancing the equations on both sides. What is the solution set? The solution set for BOTH inequalities will be ANY POINT where BOTH regions are shaded together or where BOTH shaded regions meet.Solving a system of equations means finding the values of the variables used in the set of equations. For the two examples above, we can combine both graphs and plot the area shared by the two inequalities. Graphically, it means we need to do what we just did - plot the line represented by each inequality - and then find the region of the graph that is true for BOTH inequalities. Multiple inequalities - a system of inequalitiesĪ system of inequalities has more than one inequality statement that must be satisfied.
Since that point was above our line, it should be shaded, which verifies our solution. Simplify it to \(3 \geq -1.5\) and we see that the inequality is true at the point (5,3). Example:įind all values of x and y that satisfy: \(y \geq \frac*5+6\). We have shaded the correct side of the line. In this case, that means \(0 \leq -0+10\), which is clearly true. Plug in a point not on the line, like (0,0). Since y is less than a particular value on the low-side of the axis, we will shade the region below the line to indicate that the inequality is true for all points below the line:ĥ) Verify.
In step 3 we plotted the line (the equal-to case), so now we need to account for the less-than case. Notice that it is true when y is less than or equal to.
#SOLUTIONS OF EQUATIONS AND INEQUALITIES EQUATION SYSTEMS HOW TO#
Review how to graph a line here.Ĥ) Revisit the inequality we found before as \(y \leq -x + 10\). This will form the "boundary" of the inequality - on one side of the line the condition will be true, on the other side it will not. So, \(y \leq -x + 10\) becomes \(y = -x + 10\) for the moment.ģ) Graph the line found in step 2. BUT DO NOT forget to replace the equal symbol with the original inequality symbol at the END of the problem! In doing so, you can treat the inequality like an equation. How To Solve Systems of Inequalities Graphicallyġ) Write the inequality in slope-intercept form or in the form \(y = mx + b\).įor example, if asked to solve \(x + y \leq 10\), we first re-write as \(y \leq -x + 10\).Ģ) Temporarily exchange the given inequality symbol (in this case \(\leq\)) for just equal symbol.
In light of this fact, it may be easiest to find a solution set for inequalities by solving the system graphically. There are endless solutions for inequalities. Usually this is written as = on computers because it is easier to type.
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