Achemist has two alloys, one of which is 10% gold and 15% lead and the other of which is 30% gold and 40% lead. how many grams of each of the two alloys should be used to make an alloy that contains 60 grams of gold and 88 grams of lead

Let the weight of the first alloy will be x grams.
Weight of gold will be 0.1x grams, weight of lead will be 0.15x grams.
Let the weight of the second alloy will be y grams.
Weight of gold will be 0.3y grams, weight of lead will be 0.4y grams.
Summarizing:
0.1x+0.3y=60       - gold
0.15x+0.4y=88     - lead

0.1x+0.3y=60     * 10   
0.15x+0.4y=88   * 100

x+3y=600          * 15
15x+40y=8800

15x+45y=9000      
15x+40y=8800

15x+45y=9000            /15
45y-40y=9000-8800

x+3y=600     
5y=200

x+3y=600     
y=40

y=40
x=600-3*40=480

y=40    
x=480

↑The answers.